Fundamentals Of Computational Fluid Dynamics Lomax, Pulliam

Prévia do material em texto

Fundamentals of Computational Fluid
Dynamics
Harvard Lomax and Thomas H. Pulliam
NASA Ames Research Center
David W. Zingg
University of Toronto Institute for Aerospace Studies
August 26, 1999
Contents
1 INTRODUCTION 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Problem Speci�cation and Geometry Preparation . . . . . . . 2
1.2.2 Selection of Governing Equations and Boundary Conditions . 3
1.2.3 Selection of Gridding Strategy and Numerical Method . . . . 3
1.2.4 Assessment and Interpretation of Results . . . . . . . . . . . . 4
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 CONSERVATION LAWS AND THE MODEL EQUATIONS 7
2.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Navier-Stokes and Euler Equations . . . . . . . . . . . . . . . . . 8
2.3 The Linear Convection Equation . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Di�erential Form . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Solution in Wave Space . . . . . . . . . . . . . . . . . . . . . . 13
2.4 The Di�usion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Di�erential Form . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Solution in Wave Space . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Linear Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 FINITE-DIFFERENCE APPROXIMATIONS 21
3.1 Meshes and Finite-Di�erence Notation . . . . . . . . . . . . . . . . . 21
3.2 Space Derivative Approximations . . . . . . . . . . . . . . . . . . . . 24
3.3 Finite-Di�erence Operators . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Point Di�erence Operators . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Matrix Di�erence Operators . . . . . . . . . . . . . . . . . . . 25
3.3.3 Periodic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.4 Circulant Matrices . . . . . . . . . . . . . . . . . . . . . . . . 30
iii
3.4 Constructing Di�erencing Schemes of Any Order . . . . . . . . . . . . 31
3.4.1 Taylor Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.2 Generalization of Di�erence Formulas . . . . . . . . . . . . . . 34
3.4.3 Lagrange and Hermite Interpolation Polynomials . . . . . . . 35
3.4.4 Practical Application of Pad�e Formulas . . . . . . . . . . . . . 37
3.4.5 Other Higher-Order Schemes . . . . . . . . . . . . . . . . . . . 38
3.5 Fourier Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1 Application to a Spatial Operator . . . . . . . . . . . . . . . . 39
3.6 Di�erence Operators at Boundaries . . . . . . . . . . . . . . . . . . . 43
3.6.1 The Linear Convection Equation . . . . . . . . . . . . . . . . 44
3.6.2 The Di�usion Equation . . . . . . . . . . . . . . . . . . . . . . 46
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 THE SEMI-DISCRETE APPROACH 51
4.1 Reduction of PDE's to ODE's . . . . . . . . . . . . . . . . . . . . . . 52
4.1.1 The Model ODE's . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.2 The Generic Matrix Form . . . . . . . . . . . . . . . . . . . . 53
4.2 Exact Solutions of Linear ODE's . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Eigensystems of Semi-Discrete Linear Forms . . . . . . . . . . 54
4.2.2 Single ODE's of First- and Second-Order . . . . . . . . . . . . 55
4.2.3 Coupled First-Order ODE's . . . . . . . . . . . . . . . . . . . 57
4.2.4 General Solution of Coupled ODE's with Complete Eigensystems 59
4.3 Real Space and Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Eigenvalue Spectrums for Model ODE's . . . . . . . . . . . . . 62
4.3.3 Eigenvectors of the Model Equations . . . . . . . . . . . . . . 63
4.3.4 Solutions of the Model ODE's . . . . . . . . . . . . . . . . . . 65
4.4 The Representative Equation . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 FINITE-VOLUME METHODS 71
5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Model Equations in Integral Form . . . . . . . . . . . . . . . . . . . . 73
5.2.1 The Linear Convection Equation . . . . . . . . . . . . . . . . 73
5.2.2 The Di�usion Equation . . . . . . . . . . . . . . . . . . . . . . 74
5.3 One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.1 A Second-Order Approximation to the Convection Equation . 75
5.3.2 A Fourth-Order Approximation to the Convection Equation . 77
5.3.3 A Second-Order Approximation to the Di�usion Equation . . 78
5.4 A Two-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 TIME-MARCHING METHODS FOR ODE'S 85
6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Converting Time-Marching Methods to O�E's . . . . . . . . . . . . . 87
6.3 Solution of Linear O�E's With Constant Coe�cients . . . . . . . . . 88
6.3.1 First- and Second-Order Di�erence Equations . . . . . . . . . 89
6.3.2 Special Cases of Coupled First-Order Equations . . . . . . . . 90
6.4 Solution of the Representative O�E's . . . . . . . . . . . . . . . . . 91
6.4.1 The Operational Form and its Solution . . . . . . . . . . . . . 91
6.4.2 Examples of Solutions to Time-Marching O�E's . . . . . . . . 92
6.5 The �� � Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5.1 Establishing the Relation . . . . . . . . . . . . . . . . . . . . . 93
6.5.2 The Principal �-Root . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.3 Spurious �-Roots . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.4 One-Root Time-Marching Methods . . . . . . . . . . . . . . . 96
6.6 Accuracy Measures of Time-Marching Methods . . . . . . . . . . . . 97
6.6.1 Local and Global Error Measures . . . . . . . . . . . . . . . . 97
6.6.2 Local Accuracy of the Transient Solution (er
�
; j�j ; er
!
) . . . . 98
6.6.3 Local Accuracy of the Particular Solution (er
�
) . . . . . . . . 99
6.6.4 Time Accuracy For Nonlinear Applications . . . . . . . . . . . 100
6.6.5 Global Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.7 Linear Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . . 102
6.7.1 The General Formulation . . . . . . . . . . . . . . . . . . . . . 102
6.7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.7.3 Two-Step Linear Multistep Methods . . . . . . . . . . . . . . 105
6.8 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . . . . . 106
6.9 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.10 Implementation of Implicit Methods . . . . . . . . . . . . . . . . . . . 110
6.10.1 Application to Systems of Equations . . . . . . . . . . . . . . 110
6.10.2 Application to Nonlinear Equations . . . . . . . . . . . . . . . 111
6.10.3 Local Linearization for Scalar Equations . . . . . . . . . . . . 112
6.10.4 Local Linearization for Coupled Sets of Nonlinear Equations . 115
6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7 STABILITY OF LINEAR SYSTEMS 121
7.1 Dependence on the Eigensystem . . . . . . . . . . . . . . . . . . . . . 122
7.2 Inherent Stability of ODE's . . . . . . . . . . . . . . . . . . . . . . .123
7.2.1 The Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2.2 Complete Eigensystems . . . . . . . . . . . . . . . . . . . . . . 123
7.2.3 Defective Eigensystems . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Numerical Stability of O�E 's . . . . . . . . . . . . . . . . . . . . . . 124
7.3.1 The Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3.2 Complete Eigensystems . . . . . . . . . . . . . . . . . . . . . . 125
7.3.3 Defective Eigensystems . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Time-Space Stability and Convergence of O�E's . . . . . . . . . . . . 125
7.5 Numerical Stability Concepts in the Complex �-Plane . . . . . . . . . 128
7.5.1 �-Root Traces Relative to the Unit Circle . . . . . . . . . . . 128
7.5.2 Stability for Small �t . . . . . . . . . . . . . . . . . . . . . . . 132
7.6 Numerical Stability Concepts in the Complex �h Plane . . . . . . . . 135
7.6.1 Stability for Large h. . . . . . . . . . . . . . . . . . . . . . . . 135
7.6.2 Unconditional Stability, A-Stable Methods . . . . . . . . . . . 136
7.6.3 Stability Contours in the Complex �h Plane. . . . . . . . . . . 137
7.7 Fourier Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 141
7.7.1 The Basic Procedure . . . . . . . . . . . . . . . . . . . . . . . 141
7.7.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.7.3 Relation to Circulant Matrices . . . . . . . . . . . . . . . . . . 143
7.8 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8 CHOICE OF TIME-MARCHING METHODS 149
8.1 Sti�ness De�nition for ODE's . . . . . . . . . . . . . . . . . . . . . . 149
8.1.1 Relation to �-Eigenvalues . . . . . . . . . . . . . . . . . . . . 149
8.1.2 Driving and Parasitic Eigenvalues . . . . . . . . . . . . . . . . 151
8.1.3 Sti�ness Classi�cations . . . . . . . . . . . . . . . . . . . . . . 151
8.2 Relation of Sti�ness to Space Mesh Size . . . . . . . . . . . . . . . . 152
8.3 Practical Considerations for Comparing Methods . . . . . . . . . . . 153
8.4 Comparing the E�ciency of Explicit Methods . . . . . . . . . . . . . 154
8.4.1 Imposed Constraints . . . . . . . . . . . . . . . . . . . . . . . 154
8.4.2 An Example Involving Di�usion . . . . . . . . . . . . . . . . . 154
8.4.3 An Example Involving Periodic Convection . . . . . . . . . . . 155
8.5 Coping With Sti�ness . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.5.1 Explicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.5.2 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.5.3 A Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.6 Steady Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9 RELAXATION METHODS 163
9.1 Formulation of the Model Problem . . . . . . . . . . . . . . . . . . . 164
9.1.1 Preconditioning the Basic Matrix . . . . . . . . . . . . . . . . 164
9.1.2 The Model Equations . . . . . . . . . . . . . . . . . . . . . . . 166
9.2 Classical Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.2.1 The Delta Form of an Iterative Scheme . . . . . . . . . . . . . 168
9.2.2 The Converged Solution, the Residual, and the Error . . . . . 168
9.2.3 The Classical Methods . . . . . . . . . . . . . . . . . . . . . . 169
9.3 The ODE Approach to Classical Relaxation . . . . . . . . . . . . . . 170
9.3.1 The Ordinary Di�erential Equation Formulation . . . . . . . . 170
9.3.2 ODE Form of the Classical Methods . . . . . . . . . . . . . . 172
9.4 Eigensystems of the Classical Methods . . . . . . . . . . . . . . . . . 173
9.4.1 The Point-Jacobi System . . . . . . . . . . . . . . . . . . . . . 174
9.4.2 The Gauss-Seidel System . . . . . . . . . . . . . . . . . . . . . 176
9.4.3 The SOR System . . . . . . . . . . . . . . . . . . . . . . . . . 180
9.5 Nonstationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10 MULTIGRID 191
10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.1.1 Eigenvector and Eigenvalue Identi�cation with Space Frequencies191
10.1.2 Properties of the Iterative Method . . . . . . . . . . . . . . . 192
10.2 The Basic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.3 A Two-Grid Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
11 NUMERICAL DISSIPATION 203
11.1 One-Sided First-Derivative Space Di�erencing . . . . . . . . . . . . . 204
11.2 The Modi�ed Partial Di�erential Equation . . . . . . . . . . . . . . . 205
11.3 The Lax-Wendro� Method . . . . . . . . . . . . . . . . . . . . . . . . 207
11.4 Upwind Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11.4.1 Flux-Vector Splitting . . . . . . . . . . . . . . . . . . . . . . . 210
11.4.2 Flux-Di�erence Splitting . . . . . . . . . . . . . . . . . . . . . 212
11.5 Arti�cial Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
12 SPLIT AND FACTORED FORMS 217
12.1 The Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
12.2 Factoring Physical Representations | Time Splitting . . . . . . . . . 218
12.3 Factoring Space Matrix Operators in 2{D . . . . . . . . . . . . . . . . 220
12.3.1 Mesh Indexing Convention . . . . . . . . . . . . . . . . . . . . 220
12.3.2 Data Bases and Space Vectors . . . . . . . . . . . . . . . . . . 221
12.3.3 Data Base Permutations . . . . . . . . . . . . . . . . . . . . . 221
12.3.4 Space Splitting and Factoring . . . . . . . . . . . . . . . . . . 223
12.4 Second-Order Factored Implicit Methods . . . . . . . . . . . . . . . . 226
12.5 Importance of Factored Forms in 2 and 3 Dimensions . . . . . . . . . 226
12.6 The Delta Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
12.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
13 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 233
13.1 The Representative Equation for Circulant Operators . . . . . . . . . 233
13.2 Example Analysis of Circulant Systems . . . . . . . . . . . . . . . . . 234
13.2.1 Stability Comparisons of Time-Split Methods . . . . . . . . . 234
13.2.2 Analysis of a Second-Order Time-Split Method . . . . . . . . 237
13.3 The Representative Equation for Space-Split Operators . . . . . . . . 238
13.4 Example Analysis of 2-D Model Equations . . . . . . . . . . . . . . . 242
13.4.1 The Unfactored Implicit Euler Method . . . . . . . . . . . . . 242
13.4.2 The Factored Nondelta Form of the Implicit Euler Method . . 243
13.4.3 The Factored Delta Form of the Implicit Euler Method . . . . 243
13.4.4 The Factored Delta Form of the Trapezoidal Method . . . . . 244
13.5 Example Analysis of the 3-D Model Equation . . . . . . . . . . . . . 245
13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
A USEFUL RELATIONS AND DEFINITIONS FROM LINEAR AL-
GEBRA 249
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
A.2 De�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
A.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
A.4 Eigensystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
A.5 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . 254
B SOMEPROPERTIES OF TRIDIAGONAL MATRICES 257
B.1 Standard Eigensystem for Simple Tridiagonals . . . . . . . . . . . . . 257
B.2 Generalized Eigensystem for Simple Tridiagonals . . . . . . . . . . . . 258
B.3 The Inverse of a Simple Tridiagonal . . . . . . . . . . . . . . . . . . . 259
B.4 Eigensystems of Circulant Matrices . . . . . . . . . . . . . . . . . . . 260
B.4.1 Standard Tridiagonals . . . . . . . . . . . . . . . . . . . . . . 260
B.4.2 General Circulant Systems . . . . . . . . . . . . . . . . . . . . 261
B.5 Special Cases Found From Symmetries . . . . . . . . . . . . . . . . . 262
B.6 Special Cases Involving Boundary Conditions . . . . . . . . . . . . . 263
C THEHOMOGENEOUS PROPERTYOF THE EULER EQUATIONS265
Chapter 1
INTRODUCTION
1.1 Motivation
The material in this book originated from attempts to understand and systemize nu-
merical solution techniques for the partial di�erential equations governing the physics
of 
uid 
ow. As time went on and these attempts began to crystallize, underlying
constraints on the nature of the material began to form. The principal such constraint
was the demand for uni�cation. Was there one mathematical structure which could
be used to describe the behavior and results of most numerical methods in common
use in the �eld of 
uid dynamics? Perhaps the answer is arguable, but the authors
believe the answer is a�rmative and present this book as justi�cation for that be-
lief. The mathematical structure is the theory of linear algebra and the attendant
eigenanalysis of linear systems.
The ultimate goal of the �eld of computational 
uid dynamics (CFD) is to under-
stand the physical events that occur in the 
ow of 
uids around and within designated
objects. These events are related to the action and interaction of phenomena such
as dissipation, di�usion, convection, shock waves, slip surfaces, boundary layers, and
turbulence. In the �eld of aerodynamics, all of these phenomena are governed by
the compressible Navier-Stokes equations. Many of the most important aspects of
these relations are nonlinear and, as a consequence, often have no analytic solution.
This, of course, motivates the numerical solution of the associated partial di�erential
equations. At the same time it would seem to invalidate the use of linear algebra for
the classi�cation of the numerical methods. Experience has shown that such is not
the case.
As we shall see in a later chapter, the use of numerical methods to solve partial
di�erential equations introduces an approximation that, in e�ect, can change the
form of the basic partial di�erential equations themselves. The new equations, which
1
2 CHAPTER 1. INTRODUCTION
are the ones actually being solved by the numerical process, are often referred to as
the modi�ed partial di�erential equations. Since they are not precisely the same as
the original equations, they can, and probably will, simulate the physical phenomena
listed above in ways that are not exactly the same as an exact solution to the basic
partial di�erential equation. Mathematically, these di�erences are usually referred to
as truncation errors. However, the theory associated with the numerical analysis of
uid mechanics was developed predominantly by scientists deeply interested in the
physics of 
uid 
ow and, as a consequence, these errors are often identi�ed with a
particular physical phenomenon on which they have a strong e�ect. Thus methods are
said to have a lot of \arti�cial viscosity" or said to be highly dispersive. This means
that the errors caused by the numerical approximation result in a modi�ed partial
di�erential equation having additional terms that can be identi�ed with the physics
of dissipation in the �rst case and dispersion in the second. There is nothing wrong,
of course, with identifying an error with a physical process, nor with deliberately
directing an error to a speci�c physical process, as long as the error remains in some
engineering sense \small". It is safe to say, for example, that most numerical methods
in practical use for solving the nondissipative Euler equations create a modi�ed partial
di�erential equation that produces some form of dissipation. However, if used and
interpreted properly, these methods give very useful information.
Regardless of what the numerical errors are called, if their e�ects are not thor-
oughly understood and controlled, they can lead to serious di�culties, producing
answers that represent little, if any, physical reality. This motivates studying the
concepts of stability, convergence, and consistency. On the other hand, even if the
errors are kept small enough that they can be neglected (for engineering purposes),
the resulting simulation can still be of little practical use if ine�cient or inappropriate
algorithms are used. This motivates studying the concepts of sti�ness, factorization,
and algorithm development in general. All of these concepts we hope to clarify in
this book.
1.2 Background
The �eld of computational 
uid dynamics has a broad range of applicability. Indepen-
dent of the speci�c application under study, the following sequence of steps generally
must be followed in order to obtain a satisfactory solution.
1.2.1 Problem Speci�cation and Geometry Preparation
The �rst step involves the speci�cation of the problem, including the geometry, 
ow
conditions, and the requirements of the simulation. The geometry may result from
1.2. BACKGROUND 3
measurements of an existing con�guration or may be associated with a design study.
Alternatively, in a design context, no geometry need be supplied. Instead, a set
of objectives and constraints must be speci�ed. Flow conditions might include, for
example, the Reynolds number and Mach number for the 
ow over an airfoil. The
requirements of the simulation include issues such as the level of accuracy needed, the
turnaround time required, and the solution parameters of interest. The �rst two of
these requirements are often in con
ict and compromise is necessary. As an example
of solution parameters of interest in computing the 
ow�eld about an airfoil, one may
be interested in i) the lift and pitching moment only, ii) the drag as well as the lift
and pitching moment, or iii) the details of the 
ow at some speci�c location.
1.2.2 Selection of Governing Equations and Boundary Con-
ditions
Once the problem has been speci�ed, an appropriate set of governing equations and
boundary conditions must be selected. It is generally accepted that the phenomena of
importance to the �eld of continuum 
uid dynamics are governed by the conservation
of mass, momentum, and energy. The partial di�erential equations resulting from
these conservation laws are referred to as the Navier-Stokes equations. However, in
the interest of e�ciency, it is always prudent to consider solving simpli�ed forms
of the Navier-Stokes equations when the simpli�cations retain the physics which are
essential to the goals of the simulation. Possible simpli�ed governing equations include
the potential-
ow equations, the Euler equations, and the thin-layer Navier-Stokes
equations. These may be steady or unsteady and compressible or incompressible.
Boundary types which may be encountered include solid walls, in
ow and out
ow
boundaries, periodic boundaries, symmetry boundaries, etc. The boundary conditions
which must be speci�ed depend upon the governing equations. For example, at a solid
wall, the Euler equations require 
ow tangency to be enforced, while the Navier-Stokes
equations require the no-slip condition. If necessary, physical models must be chosen
for processes which cannot be simulated within the speci�ed constraints. Turbulence
is an example of a physical process which is rarelysimulated in a practical context (at
the time of writing) and thus is often modelled. The success of a simulation depends
greatly on the engineering insight involved in selecting the governing equations and
physical models based on the problem speci�cation.
1.2.3 Selection of Gridding Strategy and Numerical Method
Next a numerical method and a strategy for dividing the 
ow domain into cells, or
elements, must be selected. We concern ourselves here only with numerical meth-
ods requiring such a tessellation of the domain, which is known as a grid, or mesh.
4 CHAPTER 1. INTRODUCTION
Many di�erent gridding strategies exist, including structured, unstructured, hybrid,
composite, and overlapping grids. Furthermore, the grid can be altered based on
the solution in an approach known as solution-adaptive gridding. The numerical
methods generally used in CFD can be classi�ed as �nite-di�erence, �nite-volume,
�nite-element, or spectral methods. The choices of a numerical method and a grid-
ding strategy are strongly interdependent. For example, the use of �nite-di�erence
methods is typically restricted to structured grids. Here again, the success of a sim-
ulation can depend on appropriate choices for the problem or class of problems of
interest.
1.2.4 Assessment and Interpretation of Results
Finally, the results of the simulation must be assessed and interpreted. This step can
require post-processing of the data, for example calculation of forces and moments,
and can be aided by sophisticated 
ow visualization tools and error estimation tech-
niques. It is critical that the magnitude of both numerical and physical-model errors
be well understood.
1.3 Overview
It should be clear that successful simulation of 
uid 
ows can involve a wide range of
issues from grid generation to turbulence modelling to the applicability of various sim-
pli�ed forms of the Navier-Stokes equations. Many of these issues are not addressed
in this book. Some of them are presented in the books by Anderson, Tannehill, and
Pletcher [1] and Hirsch [2]. Instead we focus on numerical methods, with emphasis
on �nite-di�erence and �nite-volume methods for the Euler and Navier-Stokes equa-
tions. Rather than presenting the details of the most advanced methods, which are
still evolving, we present a foundation for developing, analyzing, and understanding
such methods.
Fortunately, to develop, analyze, and understand most numerical methods used to
�nd solutions for the complete compressible Navier-Stokes equations, we can make use
of much simpler expressions, the so-called \model" equations. These model equations
isolate certain aspects of the physics contained in the complete set of equations. Hence
their numerical solution can illustrate the properties of a given numerical method
when applied to a more complicated system of equations which governs similar phys-
ical phenomena. Although the model equations are extremely simple and easy to
solve, they have been carefully selected to be representative, when used intelligently,
of di�culties and complexities that arise in realistic two- and three-dimensional 
uid
ow simulations. We believe that a thorough understanding of what happens when
1.4. NOTATION 5
numerical approximations are applied to the model equations is a major �rst step in
making con�dent and competent use of numerical approximations to the Euler and
Navier-Stokes equations. As a word of caution, however, it should be noted that,
although we can learn a great deal by studying numerical methods as applied to the
model equations and can use that information in the design and application of nu-
merical methods to practical problems, there are many aspects of practical problems
which can only be understood in the context of the complete physical systems.
1.4 Notation
The notation is generally explained as it is introduced. Bold type is reserved for real
physical vectors, such as velocity. The vector symbol~ is used for the vectors (or
column matrices) which contain the values of the dependent variable at the nodes
of a grid. Otherwise, the use of a vector consisting of a collection of scalars should
be apparent from the context and is not identi�ed by any special notation. For
example, the variable u can denote a scalar Cartesian velocity component in the Euler
and Navier-Stokes equations, a scalar quantity in the linear convection and di�usion
equations, and a vector consisting of a collection of scalars in our presentation of
hyperbolic systems. Some of the abbreviations used throughout the text are listed
and de�ned below.
PDE Partial di�erential equation
ODE Ordinary di�erential equation
O�E Ordinary di�erence equation
RHS Right-hand side
P.S. Particular solution of an ODE or system of ODE's
S.S. Fixed (time-invariant) steady-state solution
k-D k-dimensional space
�
~
bc
�
Boundary conditions, usually a vector
O(�) A term of order (i.e., proportional to) �
6 CHAPTER 1. INTRODUCTION
Chapter 2
CONSERVATION LAWS AND
THE MODEL EQUATIONS
We start out by casting our equations in the most general form, the integral conserva-
tion-law form, which is useful in understanding the concepts involved in �nite-volume
schemes. The equations are then recast into divergence form, which is natural for
�nite-di�erence schemes. The Euler and Navier-Stokes equations are brie
y discussed
in this Chapter. The main focus, though, will be on representative model equations,
in particular, the convection and di�usion equations. These equations contain many
of the salient mathematical and physical features of the full Navier-Stokes equations.
The concepts of convection and di�usion are prevalent in our development of nu-
merical methods for computational 
uid dynamics, and the recurring use of these
model equations allows us to develop a consistent framework of analysis for consis-
tency, accuracy, stability, and convergence. The model equations we study have two
properties in common. They are linear partial di�erential equations (PDE's) with
coe�cients that are constant in both space and time, and they represent phenomena
of importance to the analysis of certain aspects of 
uid dynamic problems.
2.1 Conservation Laws
Conservation laws, such as the Euler and Navier-Stokes equations and our model
equations, can be written in the following integral form:
Z
V (t
2
)
QdV �
Z
V (t
1
)
QdV +
Z
t
2
t
1
I
S(t)
n:FdSdt =
Z
t
2
t
1
Z
V (t)
PdV dt (2.1)
In this equation, Q is a vector containing the set of variables which are conserved,
e.g., mass, momentum, and energy, per unit volume. The equation is a statement of
7
8 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
the conservation of these quantities in a �nite region of space with volume V (t) and
surface area S(t) over a �nite interval of time t
2
� t
1
. In two dimensions, the region
of space, or cell, is an area A(t) bounded by a closed contour C(t). The vector n is
a unit vector normal to the surface pointing outward, F is a set of vectors, or tensor,
containing the 
ux of Q per unit area per unit time, and P is the rate of production
of Q per unit volume per unit time. If all variables are continuous in time, then Eq.
2.1 can be rewritten as
d
dt
Z
V (t)
QdV +
I
S(t)
n:FdS =
Z
V (t)
PdV (2.2)
Those methods which make various numerical approximations of the integrals in Eqs.
2.1 and 2.2 and �nd a solution for Q on that basis are referred to as �nite-volume
methods. Many of the advanced codes written for CFD applications are based on the
�nite-volume concept.
On the other hand, a partial derivative form of a conservation lawcan also be
derived. The divergence form of Eq. 2.2 is obtained by applying Gauss's theorem to
the 
ux integral, leading to
@Q
@t
+r:F = P (2.3)
where r: is the well-known divergence operator given, in Cartesian coordinates, by
r: �
 
i
@
@x
+ j
@
@y
+ k
@
@z
!
: (2.4)
and i; j, and k are unit vectors in the x; y, and z coordinate directions, respectively.
Those methods which make various approximations of the derivatives in Eq. 2.3 and
�nd a solution for Q on that basis are referred to as �nite-di�erence methods.
2.2 The Navier-Stokes and Euler Equations
The Navier-Stokes equations form a coupled system of nonlinear PDE's describing
the conservation of mass, momentum and energy for a 
uid. For a Newtonian 
uid
in one dimension, they can be written as
@Q
@t
+
@E
@x
= 0 (2.5)
with
Q =
2
6
6
6
6
6
4
�
�u
e
3
7
7
7
7
7
5
; E =
2
6
6
6
6
6
4
�u
�u
2
+ p
u(e+ p)
3
7
7
7
7
7
5
�
2
6
6
6
6
6
4
0
4
3
�
@u
@x
4
3
�u
@u
@x
+ �
@T
@x
3
7
7
7
7
7
5
(2.6)
2.2. THE NAVIER-STOKES AND EULER EQUATIONS 9
where � is the 
uid density, u is the velocity, e is the total energy per unit volume, p is
the pressure, T is the temperature, � is the coe�cient of viscosity, and � is the thermal
conductivity. The total energy e includes internal energy per unit volume �� (where
� is the internal energy per unit mass) and kinetic energy per unit volume �u
2
=2.
These equations must be supplemented by relations between � and � and the 
uid
state as well as an equation of state, such as the ideal gas law. Details can be found
in Anderson, Tannehill, and Pletcher [1] and Hirsch [2]. Note that the convective
uxes lead to �rst derivatives in space, while the viscous and heat conduction terms
involve second derivatives. This form of the equations is called conservation-law or
conservative form. Non-conservative forms can be obtained by expanding derivatives
of products using the product rule or by introducing di�erent dependent variables,
such as u and p. Although non-conservative forms of the equations are analytically
the same as the above form, they can lead to quite di�erent numerical solutions in
terms of shock strength and shock speed, for example. Thus the conservative form is
appropriate for solving 
ows with features such as shock waves.
Many 
ows of engineering interest are steady (time-invariant), or at least may be
treated as such. For such 
ows, we are often interested in the steady-state solution of
the Navier-Stokes equations, with no interest in the transient portion of the solution.
The steady solution to the one-dimensional Navier-Stokes equations must satisfy
@E
@x
= 0 (2.7)
If we neglect viscosity and heat conduction, the Euler equations are obtained. In
two-dimensional Cartesian coordinates, these can be written as
@Q
@t
+
@E
@x
+
@F
@y
= 0 (2.8)
with
Q =
2
6
6
6
4
q
1
q
2
q
3
q
4
3
7
7
7
5
=
2
6
6
6
4
�
�u
�v
e
3
7
7
7
5
; E =
2
6
6
6
4
�u
�u
2
+ p
�uv
u(e+ p)
3
7
7
7
5
; F =
2
6
6
6
4
�v
�uv
�v
2
+ p
v(e+ p)
3
7
7
7
5
(2.9)
where u and v are the Cartesian velocity components. Later on we will make use of
the following form of the Euler equations as well:
@Q
@t
+ A
@Q
@x
+B
@Q
@y
= 0 (2.10)
The matrices A =
@E
@Q
and B =
@F
@Q
are known as the 
ux Jacobians. The 
ux vectors
given above are written in terms of the primitive variables, �, u, v, and p. In order
10 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
to derive the 
ux Jacobian matrices, we must �rst write the 
ux vectors E and F in
terms of the conservative variables, q
1
, q
2
, q
3
, and q
4
, as follows:
E =
2
6
6
6
6
6
6
6
6
6
6
6
4
E
1
E
2
E
3
E
4
3
7
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
q
2
(
 � 1)q
4
+
3�
2
q
2
2
q
1
�
�1
2
q
2
3
q
1
q
3
q
2
q
1
q
4
q
2
q
1
�
�1
2
�
q
3
2
q
2
1
+
q
2
3
q
2
q
2
1
�
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(2.11)
F =
2
6
6
6
6
6
6
6
6
6
6
6
4
F
1
F
2
F
3
F
4
3
7
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
q
3
q
3
q
2
q
1
(
 � 1)q
4
+
3�
2
q
2
3
q
1
�
�1
2
q
2
2
q
1
q
4
q
3
q
1
�
�1
2
�
q
2
2
q
3
q
2
1
+
q
3
3
q
2
1
�
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(2.12)
We have assumed that the pressure satis�es p = (
 � 1)[e � �(u
2
+ v
2
)=2] from the
ideal gas law, where 
 is the ratio of speci�c heats, c
p
=c
v
. From this it follows that
the 
ux Jacobian of E can be written in terms of the conservative variables as
A =
@E
i
@q
j
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 1 0 0
a
21
(3� 
)
�
q
2
q
1
�
(1� 
)
�
q
3
q
1
�
 � 1
�
�
q
2
q
1
��
q
3
q
1
� �
q
3
q
1
� �
q
2
q
1
�
0
a
41
a
42
a
43
�
q
2
q
1
�
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(2.13)
where
a
21
=
 � 1
2
 
q
3
q
1
!
2
�
3� 
2
 
q
2
q
1
!
2
2.2. THE NAVIER-STOKES AND EULER EQUATIONS 11
a
41
= (
 � 1)
2
4
 
q
2
q
1
!
3
+
 
q
3
q
1
!
2
 
q
2
q
1
!
3
5
� 
 
q
4
q
1
! 
q
2
q
1
!
a
42
= 
 
q
4
q
1
!
�
 � 1
2
2
4
3
 
q
2
q
1
!
2
+
 
q
3
q
1
!
2
3
5
a
43
= �(
 � 1)
 
q
2
q
1
! 
q
3
q
1
!
(2.14)
and in terms of the primitive variables as
A =
2
6
6
6
6
6
6
6
6
6
6
6
4
0 1 0 0
a
21
(3� 
)u (1� 
)v (
 � 1)
�uv v u 0
a
41
a
42
a
43
u
3
7
7
7
7
7
7
7
7
7
7
7
5
(2.15)
where
a
21
=
 � 1
2
v
2
�
3� 
2
u
2
a
41
= (
 � 1)u(u
2
+ v
2
)� 
ue
�
a
42
= 
e
�
�
 � 1
2
(3u
2
+ v
2
)
a
43
= (1� 
)uv (2.16)
Derivation of the two forms of B = @F=@Q is similar. The eigenvalues of the 
ux
Jacobian matrices are purely real. This is the de�ning feature of hyperbolic systems
of PDE's, which are further discussed in Section 2.5. The homogeneous property of
the Euler equations is discussed in Appendix C.
The Navier-Stokes equations include both convective and di�usive 
uxes. This
motivates the choice of our two scalar model equations associated with the physics
of convection and di�usion. Furthermore, aspects of convective phenomena associ-ated with coupled systems of equations such as the Euler equations are important in
developing numerical methods and boundary conditions. Thus we also study linear
hyperbolic systems of PDE's.
12 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
2.3 The Linear Convection Equation
2.3.1 Di�erential Form
The simplest linear model for convection and wave propagation is the linear convection
equation given by the following PDE:
@u
@t
+ a
@u
@x
= 0 (2.17)
Here u(x; t) is a scalar quantity propagating with speed a, a real constant which may
be positive or negative. The manner in which the boundary conditions are speci�ed
separates the following two phenomena for which this equation is a model:
(1) In one type, the scalar quantity u is given on one boundary, corresponding
to a wave entering the domain through this \in
ow" boundary. No bound-
ary condition is speci�ed at the opposite side, the \out
ow" boundary. This
is consistent in terms of the well-posedness of a 1
st
-order PDE. Hence the
wave leaves the domain through the out
ow boundary without distortion or
re
ection. This type of phenomenon is referred to, simply, as the convection
problem. It represents most of the \usual" situations encountered in convect-
ing systems. Note that the left-hand boundary is the in
ow boundary when
a is positive, while the right-hand boundary is the in
ow boundary when a is
negative.
(2) In the other type, the 
ow being simulated is periodic. At any given time,
what enters on one side of the domain must be the same as that which is
leaving on the other. This is referred to as the biconvection problem. It is
the simplest to study and serves to illustrate many of the basic properties of
numerical methods applied to problems involving convection, without special
consideration of boundaries. Hence, we pay a great deal of attention to it in
the initial chapters.
Now let us consider a situation in which the initial condition is given by u(x; 0) =
u
0
(x), and the domain is in�nite. It is easy to show by substitution that the exact
solution to the linear convection equation is then
u(x; t) = u
0
(x� at) (2.18)
The initial waveform propagates unaltered with speed jaj to the right if a is positive
and to the left if a is negative. With periodic boundary conditions, the waveform
travels through one boundary and reappears at the other boundary, eventually re-
turning to its initial position. In this case, the process continues forever without any
2.3. THE LINEAR CONVECTION EQUATION 13
change in the shape of the solution. Preserving the shape of the initial condition
u
0
(x) can be a di�cult challenge for a numerical method.
2.3.2 Solution in Wave Space
We now examine the biconvection problem in more detail. Let the domain be given
by 0 � x � 2�. We restrict our attention to initial conditions in the form
u(x; 0) = f(0)e
i�x
(2.19)
where f(0) is a complex constant, and � is the wavenumber. In order to satisfy the
periodic boundary conditions, � must be an integer. It is a measure of the number of
wavelengths within the domain. With such an initial condition, the solution can be
written as
u(x; t) = f(t)e
i�x
(2.20)
where the time dependence is contained in the complex function f(t). Substituting
this solution into the linear convection equation, Eq. 2.17, we �nd that f(t) satis�es
the following ordinary di�erential equation (ODE)
df
dt
= �ia�f (2.21)
which has the solution
f(t) = f(0)e
�ia�t
(2.22)
Substituting f(t) into Eq. 2.20 gives the following solution
u(x; t) = f(0)e
i�(x�at)
= f(0)e
i(�x�!t)
(2.23)
where the frequency, !, the wavenumber, �, and the phase speed, a, are related by
! = �a (2.24)
The relation between the frequency and the wavenumber is known as the dispersion
relation. The linear relation given by Eq. 2.24 is characteristic of wave propagation
in a nondispersive medium. This means that the phase speed is the same for all
wavenumbers. As we shall see later, most numerical methods introduce some disper-
sion; that is, in a simulation, waves with di�erent wavenumbers travel at di�erent
speeds.
An arbitrary initial waveform can be produced by summing initial conditions of
the form of Eq. 2.19. For M modes, one obtains
u(x; 0) =
M
X
m=1
f
m
(0)e
i�
m
x
(2.25)
14 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
where the wavenumbers are often ordered such that �
1
� �
2
� � � � � �
M
. Since the
wave equation is linear, the solution is obtained by summing solutions of the form of
Eq. 2.23, giving
u(x; t) =
M
X
m=1
f
m
(0)e
i�
m
(x�at)
(2.26)
Dispersion and dissipation resulting from a numerical approximation will cause the
shape of the solution to change from that of the original waveform.
2.4 The Di�usion Equation
2.4.1 Di�erential Form
Di�usive 
uxes are associated with molecular motion in a continuum 
uid. A simple
linear model equation for a di�usive process is
@u
@t
= �
@
2
u
@x
2
(2.27)
where � is a positive real constant. For example, with u representing the tempera-
ture, this parabolic PDE governs the di�usion of heat in one dimension. Boundary
conditions can be periodic, Dirichlet (speci�ed u), Neumann (speci�ed @u=@x), or
mixed Dirichlet/Neumann.
In contrast to the linear convection equation, the di�usion equation has a nontrivial
steady-state solution, which is one that satis�es the governing PDE with the partial
derivative in time equal to zero. In the case of Eq. 2.27, the steady-state solution
must satisfy
@
2
u
@x
2
= 0 (2.28)
Therefore, u must vary linearly with x at steady state such that the boundary con-
ditions are satis�ed. Other steady-state solutions are obtained if a source term g(x)
is added to Eq. 2.27, as follows:
@u
@t
= �
"
@
2
u
@x
2
� g(x)
#
(2.29)
giving a steady state-solution which satis�es
@
2
u
@x
2
� g(x) = 0 (2.30)
2.4. THE DIFFUSION EQUATION 15
In two dimensions, the di�usion equation becomes
@u
@t
= �
"
@
2
u
@x
2
+
@
2
u
@y
2
� g(x; y)
#
(2.31)
where g(x; y) is again a source term. The corresponding steady equation is
@
2
u
@x
2
+
@
2
u
@y
2
� g(x; y) = 0 (2.32)
While Eq. 2.31 is parabolic, Eq. 2.32 is elliptic. The latter is known as the Poisson
equation for nonzero g, and as Laplace's equation for zero g.
2.4.2 Solution in Wave Space
We now consider a series solution to Eq. 2.27. Let the domain be given by 0 � x � �
with boundary conditions u(0) = u
a
, u(�) = u
b
. It is clear that the steady-state
solution is given by a linear function which satis�es the boundary conditions, i.e.,
h(x) = u
a
+ (u
b
� u
a
)x=�. Let the initial condition be
u(x; 0) =
M
X
m=1
f
m
(0) sin�
m
x+ h(x) (2.33)
where � must be an integer in order to satisfy the boundary conditions. A solution
of the form
u(x; t) =
M
X
m=1
f
m
(t) sin �
m
x + h(x) (2.34)
satis�es the initial and boundary conditions. Substituting this form into Eq. 2.27
gives the following ODE for f
m
:
df
m
dt
= ��
2
m
�f
m
(2.35)
and we �nd
f
m
(t) = f
m
(0)e
��
2
m
�t
(2.36)
Substituting f
m
(t) into equation 2.34, we obtain
u(x; t) =
M
X
m=1
f
m
(0)e
��
2
m
�t
sin�
m
x + h(x) (2.37)
The steady-state solution (t!1) is simply h(x). Eq. 2.37 shows that high wavenum-
ber components (large �
m
) of the solution decay more rapidly than low wavenumbercomponents, consistent with the physics of di�usion.
16 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
2.5 Linear Hyperbolic Systems
The Euler equations, Eq. 2.8, form a hyperbolic system of partial di�erential equa-
tions. Other systems of equations governing convection and wave propagation phe-
nomena, such as the Maxwell equations describing the propagation of electromagnetic
waves, are also of hyperbolic type. Many aspects of numerical methods for such sys-
tems can be understood by studying a one-dimensional constant-coe�cient linear
system of the form
@u
@t
+ A
@u
@x
= 0 (2.38)
where u = u(x; t) is a vector of length m and A is a real m � m matrix. For
conservation laws, this equation can also be written in the form
@u
@t
+
@f
@x
= 0 (2.39)
where f is the 
ux vector and A =
@f
@u
is the 
ux Jacobian matrix. The entries in the
ux Jacobian are
a
ij
=
@f
i
@u
j
(2.40)
The 
ux Jacobian for the Euler equations is derived in Section 2.2.
Such a system is hyperbolic if A is diagonalizable with real eigenvalues.
1
Thus
� = X
�1
AX (2.41)
where � is a diagonal matrix containing the eigenvalues of A, and X is the matrix
of right eigenvectors. Premultiplying Eq. 2.38 by X
�1
, postmultiplying A by the
product XX
�1
, and noting that X and X
�1
are constants, we obtain
@X
�1
u
@t
+
@
�
z }| {
X
�1
AX X
�1
u
@x
= 0 (2.42)
With w = X
�1
u, this can be rewritten as
@w
@t
+ �
@w
@x
= 0 (2.43)
When written in this manner, the equations have been decoupled into m scalar equa-
tions of the form
@w
i
@t
+ �
i
@w
i
@x
= 0 (2.44)
1
See Appendix A for a brief review of some basic relations and de�nitions from linear algebra.
2.6. PROBLEMS 17
The elements of w are known as characteristic variables. Each characteristic variable
satis�es the linear convection equation with the speed given by the corresponding
eigenvalue of A.
Based on the above, we see that a hyperbolic system in the form of Eq. 2.38 has a
solution given by the superposition of waves which can travel in either the positive or
negative directions and at varying speeds. While the scalar linear convection equation
is clearly an excellent model equation for hyperbolic systems, we must ensure that
our numerical methods are appropriate for wave speeds of arbitrary sign and possibly
widely varying magnitudes.
The one-dimensional Euler equations can also be diagonalized, leading to three
equations in the form of the linear convection equation, although they remain non-
linear, of course. The eigenvalues of the 
ux Jacobian matrix, or wave speeds, are
u; u + c, and u � c, where u is the local 
uid velocity, and c =
q
p=� is the local
speed of sound. The speed u is associated with convection of the 
uid, while u + c
and u � c are associated with sound waves. Therefore, in a supersonic 
ow, where
juj > c, all of the wave speeds have the same sign. In a subsonic 
ow, where juj < c,
wave speeds of both positive and negative sign are present, corresponding to the fact
that sound waves can travel upstream in a subsonic 
ow.
The signs of the eigenvalues of the matrix A are also important in determining
suitable boundary conditions. The characteristic variables each satisfy the linear con-
vection equation with the wave speed given by the corresponding eigenvalue. There-
fore, the boundary conditions can be speci�ed accordingly. That is, characteristic
variables associated with positive eigenvalues can be speci�ed at the left boundary,
which corresponds to in
ow for these variables. Characteristic variables associated
with negative eigenvalues can be speci�ed at the right boundary, which is the in-
ow boundary for these variables. While other boundary condition treatments are
possible, they must be consistent with this approach.
2.6 Problems
1. Show that the 1-D Euler equations can be written in terms of the primitive
variables R = [�; u; p]
T
as follows:
@R
@t
+M
@R
@x
= 0
where
M =
2
6
4
u � 0
0 u �
�1
0 
p u
3
7
5
18 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
Assume an ideal gas, p = (
 � 1)(e� �u
2
=2).
2. Find the eigenvalues and eigenvectors of the matrix M derived in question 1.
3. Derive the 
ux Jacobian matrix A = @E=@Q for the 1-D Euler equations result-
ing from the conservative variable formulation (Eq. 2.5). Find its eigenvalues
and compare with those obtained in question 2.
4. Show that the two matricesM and A derived in questions 1 and 3, respectively,
are related by a similarity transform. (Hint: make use of the matrix S =
@Q=@R.)
5. Write the 2-D di�usion equation, Eq. 2.31, in the form of Eq. 2.2.
6. Given the initial condition u(x; 0) = sinx de�ned on 0 � x � 2�, write it in the
form of Eq. 2.25, that is, �nd the necessary values of f
m
(0). (Hint: use M = 2
with �
1
= 1 and �
2
= �1.) Next consider the same initial condition de�ned
only at x = 2�j=4, j = 0; 1; 2; 3. Find the values of f
m
(0) required to reproduce
the initial condition at these discrete points using M = 4 with �
m
= m� 1.
7. Plot the �rst three basis functions used in constructing the exact solution to
the di�usion equation in Section 2.4.2. Next consider a solution with boundary
conditions u
a
= u
b
= 0, and initial conditions from Eq. 2.33 with f
m
(0) = 1
for 1 � m � 3, f
m
(0) = 0 for m > 3. Plot the initial condition on the domain
0 � x � �. Plot the solution at t = 1 with � = 1.
8. Write the classical wave equation @
2
u=@t
2
= c
2
@
2
u=@x
2
as a �rst-order system,
i.e., in the form
@U
@t
+ A
@U
@x
= 0
where U = [@u=@x; @u=@t]
T
. Find the eigenvalues and eigenvectors of A.
9. The Cauchy-Riemann equations are formed from the coupling of the steady
compressible continuity (conservation of mass) equation
@�u
@x
+
@�v
@y
= 0
and the vorticity de�nition
! = �
@v
@x
+
@u
@y
= 0
2.6. PROBLEMS 19
where ! = 0 for irrotational 
ow. For isentropic and homenthalpic 
ow, the
system is closed by the relation
� =
�
1�
 � 1
2
�
u
2
+ v
2
� 1
�
�
1
�1
Note that the variables have been nondimensionalized. Combining the two
PDE's, we have
@f(q)
@x
+
@g(q)
@y
= 0
where
q =
�
u
v
�
; f =
�
��u
v
�
; g =
�
��v
�u
�
One approach to solving these equations is to add a time-dependent term and
�nd the steady solution of the following equation:
@q
@t
+
@f
@x
+
@g
@y
= 0
(a) Find the 
ux Jacobians of f and g with respect to q.
(b) Determine the eigenvalues of the 
ux Jacobians.
(c) Determine the conditions (in terms of � and u) under which the system is
hyperbolic, i.e., has real eigenvalues.
(d) Are the above 
uxes homogeneous? (See Appendix C.)
20 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
���
� � ��� �
	�� 
� ��� ��� � ��� ��� � � � � �
�
�
� � �
� ff fi �
�
� �flff � ffi
�! fl"$#&%�%'#& )(+*-,/.
,/.10�03254768,9*:#& 1;=<&#?>&0$@9 1*- 1<A41 1;B,/0C68DFE
G141*:DHG1#I(KJL#&41@M%'#ND103OP03254768,9*:#& F;
"3#� 5,/6�*: MQ76�@R,9*S6�OND10$@9*->868,9*->�03;L(+*T,/.U@R03;9QV03"W,X,/#ZYV#�,/.[;RQ76�"$0+6� 1DU,/*-%�08\^]Z 10_"C6� `6�QFQ1@9#Cab*:%�68,/0
,/.F03;90c;9*:%M41O-,/6� 103#�41;9OTE'6� 1Dd,9.103 e;R#&O->�0c,/.10f@90$;941OT,/*: F<MD1*Tgh0$@903 F"30Z0$2N4168,/*-#& 1;3\jifO-,903@9 168,/*T>&03OTE&J
#& 10M"C6� k6�QFQ1@9#Cab*:%�68,/0K,/.F0U;9Q76l,/*S68OmD10$@9*T>�6l,/*->�03;cn1@9;R,3Jh,/.F03@90$YoE�Q1@R#bD14F"3*:F<e6d;REN;B,/03%p#�qX#&@Br
D1*- 76�@RE`DF*-gs03@90$ o,/*S68OV0$254768,/*-#& 1;$\jtu.10f,9*:%'0fD10$@9*T>�6l,/*->�03;v6�@90c6�Q1Q1@R#CaF*-%d6l,/03D� 10WaN,3JFO:036�D1*- 1<w,9#
6kxzy|{d}W~!{'�€/ƒ‚Ny|„N…†{d}Wx|‚7‡3ˆ'(+.1*-"ƒ.HQ1@R#bD14F"303;M6‰;R0$,M#�qŁD1*Tgh0$@90$ 1"30'03254768,9*:#& F;w(+.1*-"ƒ.fl"C68 HYV0
;9#�O->&0$D‹\Łtu.1*-;+*:;u,/.10=6�Q1QF@9#o6�"ƒ.†03%'Q1.76�;R*:Œ$03D�.F03@908\Ł�! ,/.1*-;f"ƒ.768QF,/0$@3J7,/.10w"3#� 1"30$QF,Z#�qmn7 1*T,/0Wr
D1*Tgh0$@903 F"30Ž68Q1Q1@9#Cab*:%�68,9*:#& 1;Z,9#�Q768@R,/*:6�OjD10$@9*->868,9*->�03;M*-;KQ1@90$;903 o,903D‹\†tu.103;R0Ž"C6� HYs0�6�Q1Q1O-*:0$D
03*T,/.10$@f,9#d;9Q168,/*:6�OmDF03@9*T>868,/*T>&03;f#&@+,/*:%'0wD103@R*->868,/*T>&0$;3\c]4F@c03%'Q1.768;9*:;f*: ‰,/.1*-;_.76�Qb,/03@c*:;‘#& 
;9Q168,/*:6�OhD10$@9*T>�6l,/*->�03;$’F,/*-%�0fD103@R*->868,/*T>&0$;u6�@90Z,9@90368,/0$De*: ‰_.76�Qb,/03@+“b\P”N,/@968,/0$<&*:0$;_q•#&@u6�QFQ1O-EN*: F<
,/.F03;90Kn7 1*T,/0–r!D1*Tgh0$@903 F"30K6�Q1QF@9#Cab*:%�68,/*-#& 1;_(+*-O:OhYV0KD1*:;R"341;R;90$D�*: ‰_.768QF,/0$@f—1\
icO-OL#�qj,9.10[%d6l,/03@R*S6�O˜YV03O-#I(�*-;ZQ1@90$;90$ 5,903D)*: A6Ł68@R,/0$;9*:6� A;REN;R,903%‰\w™
0U0$%�Q1.16�;9*-Œ30U,/.F0
qš6�"W,‘,9.768,+2541*T,/0K<&03 F03@/68O›"3OS68;9;90$;f#8qL%'03;R.103;‘0WabQ1@R03;9;R03DA*- ‰<�03 10$@/6�O›"$41@R>N*:O-*: 1036�@Ł"$#b#&@RD1*: 168,/0$;
*: Aœ7‚N?žWyŸ9� Lž!œFNƒ}"C68 HYV0',9@/6� 1;Bq•#&@9%'03D),/#†6†41 F*-q•#&@R%¡Ł6�@R,903;R*S6� 
%�0$;9.
(+*T,/.
032541*:;RQ76�"$03D
*: o,903@R>86�O-;‹*: K6u;R#8r!"36�O:O-03D�ƒ‡l{cœV¢Nx£�xšyŸ‡l„V� 8ž!œFNƒ}WJI6�;›;R.1#?(+ M*- K¤L*:<&4F@90^¥b\-¦&\§tu.10Zƒ‡l{cœV¢Nx£�xšyŸ‡l„V� 
ž!œFNƒ}h*:;˜41 1*Tq•#&@9%‰’�68O:O�,9.10v<�03#&%'0$,9@9*:"^>86�@R*S68,9*:#& *-;§6�Y1;9#�@9YV03DM*- o,/#+>�68@9*S68Y1O:0^"$#b0W¨'"3*:0$ o,/;L#8qF,/.F0
,/@96� 1;Bq•#&@9%'03D)03254768,9*:#& F;3\e¤7#&@,/.F*:;w@90C68;9#& ‹J§*: )%U41"ƒ.
#�qŁ,9.10'q•#�O:O:#?(+*- 1<6�"3"$41@/68"$E
6� 76�OTEN;9*:;
(v0�41;R0�6� ©032541*:;RQ76�"$03DªŁ6�@B,/0$;9*S68 «;REN;B,/03% (+*-,9.1#&4F,'YV03*- 1<H41 FD141O-E¬@R03;B,/@9*-"$,9*->&0‰#&@'O:#&;R*: 1<
Q1@96�"$,9*:"C68O›6�Q1Q1O-*:"368,/*-#& ‹\
­®?¯ ° ±²^³�±²µ´M¶A· ¸º¹C¶A¹$»L±f¼h½¾¹$¿fl±Àj±¶AÁ‘± ÂÄÃ`»L´=»^¹3Ã'¶
tu.10;R*:%'Q1O:0$;R,Ł%'03;R.*: o>&#&OT>N*: 1<UYV#�,9.Ž,/*:%'0Z68 1De;9Q768"30K*:;v;9.1#?(+ ‰*: ¤L*-<&41@90c¥F\
�
\jic e*- 1;9QV03"–r
,/*-#& Å#�q‘,/.1*-;Un7<&41@R0Qs0$@9%'*-,/;M41;[,/#kD10$n1 10‰,9.10e,/0$@9%';`6� 1D¬ 1#�,/68,/*-#& Å 1030$D103DÆ,/#kD10$;9"3@R*:YV0
�
¦
�&� Ç^ȑÉZÊcËmÌXÍÏÎ5ЫÑLÒÔӑÒ3ËmÌPÕzÖZÒÔÑ^Ñ^ÌX͑ÌPÓMÇ^Ì«ÉZÊXÊX͏×jØZÒBنÉc˧ÒW×uÓZÚ
x
y
¤L*:<�41@90K¥F\T¦&Û Üj.oEN;9*:"36�O˜6� 1D‰"3#�%�Q14b,ƒ68,9*:#& 768Oh;9Q16�"30$;3\
n7 1*T,/0ZDF*-gs03@90$ 1"30K68Q1Q1@9#Cab*:%�68,9*:#& 1;$\^�! <&03 F03@/68OŸJb,9.10D10$Qs0$ 1D103 o,+>86�@9*:6�Y1O-03;3JbݘJbq•#&@Ł0–aF6�%�QFO:0�J
6�@R0‘q•41 F"$,/*-#& 1;v#�q‹,/.F0c*: FD103QV03 1DF03 o,Ł>86�@9*:6�Y1O-03;jޖJF68 1Ddߘà/áVàƒâF\j¤7#&@j,9.10fn7@R;R,_;90$>�03@96�Oh"ƒ.768QF,/0$@9;
(v0["3#& 1;R*:D10$@Q1@9*-%d68@9*:OTEd,9.10e¦$rÔãÏ"36�;90[ÝAäåÝmæ•ß˜à9ÞRç–\K™ª.F03 k#& 1OTE�#& F0M>86�@9*:6�Y1O:0w*:;cD103 1#8,/03DhJ
D10$Qs0$ 1D103 F"30U#& ,/.F0w#8,/.10$@f*-;+6�;9;R41%�0$D‹\_tu.F0=%'03;R.�*: FD10Waeq•#�@+ß�*:;w� éèP�?žXê7Jë6� 1D‰,/.768,uq•#�@
ÞŁ*-;w� éèP�Iž_ìL\jtu.F03 �#� �6� 0$2N4F*:;9Q16�"30$D�<&@R*:D
ß ä ßNíuä«ê5îUß æš¥F\-¦?ç
Þ¡ä Þ!ï=äðì‹îUÞPäªì›ñ 暥F\
�
ç
(+.10$@90[îUßk*:;u,9.10M;RQ76�"3*- 1<d*- †ß)6� FD�îUÞu,/.10=;9Q768"3*: F<d*: Þ–Js6�;c;R.1#?(+ A*: †¤L*:<&4F@90w¥F\
�
\‘òf#�,90
,/.168,cñeäóî[Þ_,/.1@R#&41<&.1#�4F,C\vô˜68,/0$@Zõ6� 1D†öj6�@90K41;R03D�q•#�@‘áŽ6� 1D†â�*: †6`;9*:%'*:O:6�@v(_6CE&\_™ª.F03 
ìLà£ê&à–õsàƒö6�@R0†4F;903D©q•#&@�#8,/.10$@�Q14F@9QV#&;90$;Aæ•(+.F*:"ƒ.º*-;�;R#&%�0W,/*-%�0$;' F03"30$;9;96�@REFçWJ+O-#b"36�Ou"3#& o,90WaN,
;9.F#&41O:D%�6�÷�0,9.10K%�036� 1*: F<[#&Yo>b*-#&41;$\
tu.10w"3#& o>&0$ o,/*:#� †q•#&@u;R41Y1;9"$@9*-QF,f6� 1D‰;941QV03@R;9"3@R*:QF,‘*- 1D10Wab*- 1<`*:;+6�;uq•#&O-O:#?(+;3Û
Ýmæ•Þ‹ø«õFñVç¡ä ÝmæRùúì'ø©õNû|ñëçXäðÝ7ïCü1ý
ÝLæ|ß[ø¬þîUßsç¾ä ÝmæRù êcøÆþŽû:î[ßsçXäßÝNíRü�� 暥F\ ¥&ç
Ýmæ|ß[ø¬þî[ߘàRޘø«õFñVç¡ä Ý��
ï?ü1ý��
íRü��
òf#8,/*:"$0Z,/.76l,u(+.103 ‰41;90$D�6�O-#& 108JFYV#�,/.e,/.F0K,9*:%'06� 1D;RQ76�"30w*: FD1*:"$03;u6�QFQs036�@‘6�;+6[;941Y1;R"3@R*:QF,3J
Y14F,[(+.F03 Æ41;90$D¬,9#&<&0W,/.10$@3Jj,/*-%�0Ž*-;[6�OT(Ł6CEN;`6�;941QV03@R;9"$@9*:Qb,'6� 1D¬*:;U4F;9476�O-O-E
03 1"$O:#&;R03DÅ(+*-,9.
Q76�@R03 o,/.F03;90$;f,9#'DF*:;R,9*: 1<�41*:;R.‰*T,Łq•@9#�% 6� ‰0WabQV#& 103 o,3\
ÎNÐ��&кنÌvÚbÈcÌPډÉcÓcÖ�ÑLÒÔÓfÒ3ËmÌX՟ÖZÒÔÑLÑ^ÌXÍfÌXÓUÇ^Ì ÓM×=Ë1Éc˧ÒW×uÓ �
¥
t�
xj-2	 j-1 j j+1	 j+2	
n
n-1
n+1
x
t
Grid or
Node
Points
∆
∆
¤L*-<&41@R0¥F\
�
Û ”bQ76�"$0WrŸ,/*:%'0<&@R*:D6�@R@/6� 1<�03%'03 o,C\
ãZ03@R*->868,/*T>&0$;�68@900WabQ1@R03;9;R03D 68"3"3#�@9D1*- 1<),9#A,/.1041;R476�O+"3#& o>�03 o,/*-#& 1;3\©tu.541;�q•#&@[Q16�@R,9*S6�O
D10$@9*->868,9*->�03;+*: e;RQ76�"30w#&@Ł,9*:%'0c(_0w41;R0=*- 5,903@R"ƒ.76� 1<&036�Y1OTE
��
Ýeä
Ý
ß
à
�
Ýeä
Ý
Þ
à
����
Ýeä
��
Ý
ß
�
à 0$,9"�\ 暥F\ —oç
¤7#&@Ł,/.F0w#�@9D1*- 76�@RE�,/*-%�0ZDF03@9*T>868,/*T>&0*: e,9.10K;R,/4FDFE#�q^]Kã���� ;u(_0K41;R0
��b�
Ý
�
Þ
暥F\���ç
�! �,/.F*:;+,/0WaN,;R41Y1;R"3@9*-QF,/;#� �D103QV03 FD103 o,>86�@9*:6�Y1O-03;f6�@90= 10$>�03@41;R03D�,/#d0WabQ1@90$;9;KD10$@9*->868,9*->�03;3\
tu.541;‘Ý
�
(+*-O:Oh F#�,‘Ys041;R03D‰,/#`@90$Q1@90$;903 o,f,/.10n7@R;R,‘D103@R*->868,9*->&0#8qL݉(+*-,/.‰@90$;9QV03"W,‘,9#'ߘ\
tu.10c F#�,ƒ68,9*:#& [q•#&@XD1*Tgh0$@90$ 1"30c6�Q1Q1@R#?ab*-%d68,9*:#& F;^q•#&O:O-#I(+;^,9.10‘;/6�%'0fQF.1*:O-#&;9#&QF.5EUY14F,Zæ•(+*-,9.
#& 10P0Wab"$03QF,9*:#& ëç›*-,§*-;§ F#�,L4F 1*:254108\ff�vE,9.1*:;›(v0_%'0C6� K,/.76l,m,9.10v;BEN%UYs#�Oflfiu*-;§4F;903D=,/#‘@903QF@903;R03 o,
6[D1*-gs03@R03 1"$0M68Q1Q1@9#Cab*:%�68,9*:#& d,9#'6`DF03@9*T>868,/*T>&0K;941"ƒ.‰,/.76l,CJFq•#&@u0–aF6�%�QFO:0�J
fi
��ffi 
!�
à fi
����ffi 
����
暥F\ “&ç
Y14F,u,9.10KQ1@90$"3*-;90K 768,941@90'æš6� 1D‰#&@9D10$@ƒç_#�q§,/.F0K6�Q1Q1@R#?ab*-%d68,9*:#& d*:;u 1#�,+"36�@9@R*:0$D‰*- Ž,/.10K;BEb%MYs#&O
fi5\U]c,/.F03@Z(_6?EN;w6�@R0[41;90$Dk,9#D10$,903@9%'*: F0U*-,9;ZQ1@90$"3*-;90[%'0C6� 1*- 1<1\Ktu.10[#& F0[0Wab"30$QF,/*-#& k*:;Z,/.F0
;REN%UYV#&O˜îeJ1(+.1*-"ƒ.†*-;uD10$n7 F03D�;R41"ƒ.†,9.768,
îUÞ!ï=ä Þ!ï?ü#"%$HÞ!ï1à îUßNí+äªßNíRü#"ff$HßNíIà î[Ýëï=äªÝ7ï?ü#"&$ÅÝ7ïbà 0$,9"�\ 暥F\�'�ç
™ª.10$ ‰,9.103@R0=*-;u 1#`;941Y1;R"3@R*:QF,f#& ‰îUÞŁ#&@+îUߘJ7,/.10K;RQ76�"3*- 1<`*:;u41 F*-q•#&@R%†\
�
—
Ç^ȑÉZÊcËmÌXÍÏÎ5ЫÑLÒÔӑÒ3ËmÌPÕzÖZÒÔÑ^Ñ^ÌX͑ÌPÓMÇ^Ì«ÉZÊXÊX͏×jØZÒBنÉc˧ÒW×uÓZÚ
­®)( *,+�´MÁ‘± ½ ±ÀX¹.-w´w»j¹.-'± /0+1+AÀjÃ32†¹54 ´=»j¹$Ã�¶A²
iÆD1*-gs03@R03 1"$0Ł6�Q1QF@9#Cab*:%�68,/*-#& K"C6� UYV0P<&03 10$@/68,903DU#�@m0W>�68O:4768,903D=Y5Ew%'0C6� F;L#�që6‘;9*-%�Q1O-0jtL6CENO:#�@
;90$@9*-03;Ž0–aFQ16� 1;9*-#& ‹\ ¤ë#�@�0–a168%�Q1O-0�Ju"$#& 1;9*-D103@�Ýmæ•ß˜à9ÞRç[(+*-,/.©Þ[nFab03D‹\ótu.F03 ‹Juq•#&O-O:#?(+*: 1<k,/.F0
 1#�,/68,/*-#& A"3#� 5>�03 o,/*-#& )<&*->�03 k*: 6�j25;3\˜¥F\T¦U,/#‰¥F\ ¥FJ‹ß
ä ê5îUßfl6� FD)Ýmæ|ß�øðõbî[ßsçcä ÝmæTêNîUß�ø
õFîUßsçXäßÝNí9ü1ýI\7�LaFQ16� 1D1*- 1<',9.10KOS68,R,/0$@_,/0$@9%µ6�YV#&4F,uß�<&*->�03;
ÝNí9ü1ý+äðÝbíLø æzõFîUßsç�8
Ý
ß:9
í
ø
¦
�
æzõFîUßsç
�
8
��
Ý
ß
�
9
í
ø<;=;=;?ø
¦
ì%>
æŸõFîUßsç
ï
8
ï
Ý
ß
ï
9
í
ø<;=;=;m暥F\@?&ç
ô›#b"36�OoD1*Tgh0$@903 F"30P6�Q1Q1@R#CaF*-%d6l,/*:#� 1;‹,/#‘6+<&*->�03 wQ76�@R,9*S6�O�D103@R*->868,/*T>&0X"C68 =YV0jq•#&@9%'03DKq•@9#&% O:*: F0C6�@
"3#�%UY1*: 168,/*-#& 1;Ł#�qmÝNíZ6� 1DÝNí9ü1ýfq•#&@‘õ�äBA`¦&à�A
�
à5;=;=;T\
¤7#&@u0WaF6�%'Q1O:08J1"3#� 1;9*-D103@u,/.F0wtm6CENO:#&@u;R03@9*-03;‘0WabQ76� F;9*:#� ‰q•#&@uÝNíRü#"ƒÛ
ÝNíRü#"jäªÝNí^ø æšîUßsç�8
Ý
�9
í
ø
¦
�
æšî[ßsç
�
8
�
Ý
ß
�
9
í
øC;=;=;Iø
¦
ì&>
æšîUßsç
ï
8
ï
Ý
ß
ï
9
í
ø<;=;=; 暥F\@D&ç
òf#?(;941YF,9@/6�"W,‘Ýbíc6� 1D‰D1*->N*:DF0Y5Eeî[߆,/#'#&YF,/6�*: 
ÝbíRü#"&$flÝNí
î[ß
äE8
Ý
ßF9
í
ø
¦
�
æšîUßsç�8
�
Ý
ß
�
9
í
ø<;=;=; æz¥F\T¦
�
ç
tu.541;L,9.10Ł0WabQ1@R03;R;9*:#� ‰æšÝNí9ü#"G$=ÝNí3çIH�îUß�*-;L6f@90C68;9#& 768Y1O:0v6�Q1Q1@R#CaF*-%d6l,/*:#� wq•#&@�JLKNM
K
��O
í
6�;LO-#& 1<Z68;
îUßA*:;u;R%d68O:OŸ\Xò‘0WaN,f"3#& 1;R*:D10$@‘,9.10K;9Q768"30wD1*-gs03@R03 1"$0M6�QFQ1@9#Cab*:%�68,/*-#& )æšÝNíRü#"%$ÅÝNí�P�"BçQHFæ
�
îUßsçW\
�LabQ76� 1Dð,/.10A,/0$@9%';*: ,/.10k 541%'03@968,/#&@‰6�YV#&4F,'ê«6� 1Dð@903<�@9#&41Qª,/.F0)@R03;R41O-,e,9#Æq•#&@R% ,/.F0
q•#&O-O:#?(+*: 1<M03254768,9*:#& 
ÝNí9ü#"R$flÝNí�P�"
�
îUß
$S8
Ý
ß
9
í
ä
¦
“
î[ß
�
8
�T
Ý
ß
T
9
í
ø
¦
¦
�8�
îUß�U�8
�V
Ý
ß
V
9
í&WNWNW
æz¥F\T¦&¦Iç
™ª.10$ �0WabQ1@R03;R;903DA*- e,/.1*-;‘%�6� 1 F03@3J7*T,+*:;+"3O-0C6�@Ł,9.768,+,/.F0=D1*-;9"$@90$,90,/03@R%�;+#& ,/.F0=O-0$q|,+;9*-D10K#�q
,/.F0f0$2N4168,/*-#& �@R03Q1@R03;R03 o,u6n7@9;B,PD103@R*->868,/*T>&0u(+*-,9.d6K"30$@R,/6�*: �6�%'#&41 o,X#�q‹0$@9@R#&@j(+.1*:"ƒ.d6�Q1QV0C6�@R;
#& ¬,/.10@R*:<&.o,[;R*:D10#�q‘,/.100$2N416�O+;9*:<� ‹\Å�£,'*:;`6�O-;9#k"3O-0C6�@U,9.768,[,/.F0‰0$@9@R#&@`D103QV03 1DF;�#& ¬,/.F0
<&@R*:DH;RQ76�"$*: 1<�,9#�6�"$03@B,ƒ6�*- H#&@RD103@$\�tu.10Ž0$@9@R#&@w,903@R%¾"$#& o,ƒ6�*- 1*: 1<,/.F0d<&@R*:DH;RQ76�"$*: 1<�,9#�,/.F0
O:#?(v03;B,jQs#?(v03@j<&*T>&0$;L,/.10u#�@9D10$@L#�që,/.F0Ł%�0W,/.1#ND‹\L¤ë@R#&%X�j2V\&¥F\-¦
�
J8(v0u;90$0Ł,/.768,m,/.10u0–abQ1@90$;9;9*-#& 
æšÝNíRü#"!$'Ýbí$çQH8î[ß�*:;X6cn1@9;R,Br!#&@RD103@X68Q1Q1@9#Cab*:%�68,9*:#& M,9#YJZKNM
K
�
í
O
\^”b*:%'*:O:6�@9OTE&J[�j2ë\5¥F\-¦�¦u;9.1#?(+;j,/.76l,
æšÝNíRü#"�$kÝNí)P�"ÔçQH1æ
�
îUßsçv*:;_6M;90$"3#& 1DNr!#&@RD103@+6�QFQ1@9#Cab*:%�68,/*-#& �,9#[6Mn7@R;R,uD10$@9*T>�6l,/*->�0�\^tu.F0OS68,R,/03@
*:;X@R0$q•03@R@90$D�,/#=6�;X,/.10+,/.F@9030–r!QV#&*: o,X"$03 o,/0$@903DeDF*-gs03@90$ 1"30f6�Q1QF@9#Cab*:%�68,/*-#& ‹Jo6� FD�#& F0f#�q|,903 ';90$03;
,/.F0w;R41%'%d6�@BEŽ@R03;94FO-,+Q1@R03;90$ o,/03D�*: e,/.10q•#&@R%
8
Ý
ß
9
í
ä
ÝbíRü#"R$ÅÝNí)P�"
�
îUß
ø]\dæzîUß
�
ç æz¥F\T¦
�
ç
ÎNÐ ÎNЫÑmÒÔÓfÒ3ËmÌX՟ÖZÒÔÑ^ÑLÌP͑ÌXÓUÇ^Ì ×uÊXÌX͑ÉcË_×uÍfÚ �
�
­®ƒ­ ¸º¹C¶A¹W»^±f¼h½ ¹$¿Å±Àj±¶AÁ+± ^_+H±ZÀX´w»^Ã'ÀX²
`�ab`�aIc dfehgIikjml gonqpsr�pRiutvpxwzy{psr�|%j�ehr�}
Üm0$@9.768Q1;c,/.F0=%'#&;R,‘"$#&%�%'#& ‰0WaF6�%'Q1O:0$;f#8q^n7 1*T,/0Wr£D1*-gs03@R03 1"$0q•#&@9%U4FOS6�;‘6�@R0w,9.10=,9.1@90$0Wr!QV#&*- 5,
"30$ o,/03@R03Dbr£D1*-gs03@R03 1"$0U6�QFQ1@9#Cab*:%�68,/*-#& 1;_q•#�@Ł,/.10n7@R;R,f6� 1D;90$"3#& FD�D10$@9*->868,9*->�03;3Û
"
8
Ý
ß 9
í
ä
¦
�
îUß
æšÝNíRü#"%$HÝbí�P�"Bç›ø~\ŽæzîUß
�
ç æz¥F\T¦?¥oç
8
��
Ý
ß
�
9
í
ä
¦
îUß
�
æ•ÝbíRü#"&$
�
ÝNí^ø¬Ýbí�P�"Bç›ø~\ŽæzîUß
�
ç æz¥F\T¦C—5ç
tu.10$;906�@R0f,9.10cY768;9*:;Xq•#&@jœ7‡ly|„7xXˆ�y Z}W€/}W„hƒ}‡ƒœ7}W€R�xԇl€/žX;R*: 1"$0f,/.F0$E'<&*->�0Z6� d6�Q1Q1@R#CaF*-%d6l,/*:#� `,9#
6=DF03@9*T>868,/*T>&0f68,_#� 10cD1*-;9"$@90$,90cQV#&*: o,v*: �6=%'03;R.Ž*: ',/03@R%�;P#�q›;R41@9@R#&41 1DF*: 1<=Qs#&*- o,/;3\€‘#?(_0W>&03@$J
 10$*-,/.F03@w#�q_,9.103;R0d0WabQ1@R03;R;9*:#� 1;=,903O:Oj4F;=.1#?( #8,/.10$@=QV#&*: o,9;w*- ),9.10'%�0$;9.fl68@90'D1*-gs03@R03 1"$03DH#�@
.1#?( YV#&41 1D768@RE"3#� 1D1*-,9*:#& F;c6�@R0=0$ Fq•#&@R"303Dh\c”N41"ƒ.A6�D1DF*-,/*-#& 76�O›*- Fq•#&@R%d68,9*:#& e@R032541*-@903;c6�%'#&@90
;9#�Q1.1*:;B,/*-"C68,903Deq•#&@9%M41OS68,9*:#& h\
`�ab`�ao‚ ƒ |%j�r�go„…lBg†n{pRr�pRiutvpxwmy‡pRr�|%j�e7r�}
_#& 1;R*:D10$@u,/.10K@90$OS68,9*:#& 
æˆfi
���
ÝsçzíŁä
¦
îUß
�
æšÝNí9ü#"&$
�
ÝbíLøÆÝNí)P�"Bç æz¥F\T¦G�&ç
(+.1*-"ƒ.e*:;u6MQV#&*- 5,_D1*-gs03@R03 1"$06�Q1QF@9#Cab*:%�68,/*-#& �,9#[6U;90$"3#& 1DD10$@9*T>�6l,/*->�0�\jò‘#?(ßO-0$,_41;uD103@R*->�06
{'�xš€WyŁ‰w#&QV03@/6l,/#&@+@90$Q1@90$;90$ 5,/68,/*-#& †q•#�@+,9.10K;/6�%'0=6�Q1Q1@R#?ab*-%d68,9*:#& h\P_#& 1;9*-D103@+,/.F0wq•#&4F@‘QV#&*- 5,
%'03;9.�(+*T,/.�YV#&41 FD76�@RE†QV#&*: o,/;Z68,�‹‰6� 1D6ŒK;9.1#?(+ kYV03O:#?(K\ò‘#�,/*-"30M,9.768,f(+.103 A(v0[;9QV0C6�÷‰#�q

,9.10= 541%MYs0$@‘#8q^QV#&*: o,/;+*- �6`%�0$;9.ŽFJë(v0w%�036� ‰,9.10w N4F%UYs0$@f#8qŁy|„7x!}W€WyŸ‡l€_Qs#&*- o,/;+0Wab"3O-41D1*- 1<
,/.F0wYV#&41 FD76�@9*-03;$\
‹ ¦
�
¥ — Œ
ßeä
�
$ $ $ $ 
ê[ä ¦ ; ; ‘
¤ë#�41@uQs#�*: o,u%'03;9.h\vîU߉䒐RH1擑 øª¦Iç
òf#?( *-%�QV#&;R0ŽãZ*:@R*:"ƒ.1O-0$,=Ys#�41 1D76�@BEÅ"3#� 1D1*-,9*:#& F;3JjÝmæ
�
ç[ä ݕ”Ià[Ým斐˜ç'ä ݕ—K68 1D¬41;90e,/.F0
"30$ o,/03@R03DAD1*Tgh0$@90$ 1"30=6�Q1Q1@R#CaF*-%d6l,/*:#� e<&*->�03 �YoE˜�j2ë\s¥b\-¦G�`68,‘0$>�03@BE�QV#&*: o,‘*: ,/.F0=%'03;R.‹\_™
0
™oš‡›œvž@Ÿ@Ÿ¡ =›£¢¤ž@¥¦›%§“¨G›©“›Iª£«¦¬[ � =›I¢“ž@¥�­�§“ž@¥¦›ff«.®Z›I¢¤­)§“«¦¢�©“¨G«¦¢“§“Ÿ@¯¦°
�
“
Ç^ȑÉZÊcËmÌXÍÏÎ5ЫÑLÒÔӑÒ3ËmÌPÕzÖZÒÔÑ^Ñ^ÌX͑ÌPÓMÇ^Ì«ÉZÊXÊX͏×jØZÒBنÉc˧ÒW×uÓZÚ
6�@R@9*->�0K68,+,/.10q•#&4F@u03254768,9*:#& 1;$Û
æ“fi
���
Ýsç
"
ä
¦
îUß
�
æšÝ�”k$
�
Ýv"‹ø¬Ý
�
ç
æ“fi
���
Ýsç
�
ä
¦
îUß
�
æšÝ�"&$
�
Ý
�
øÆÝ
T
ç
æ“fi
���
Ýsç
T
ä
¦
îUß
�
æšÝ
�
$
�
Ý
T
øÆÝ
U
ç
æ“fi
���
Ýsç
U
ä
¦
îUß
�
æšÝ
T
$
�
Ý
U
øÆÝ�—!ç æz¥F\T¦?“oç
™
@9*-,9*: 1<U,9.103;R0K03254768,9*:#& 1;+*- ,9.10K%�#�@90;94F<&<&03;B,/*T>&0Kq•#&@9%
æ“fi
���
Ýhç
"
ä æµÝ�” $
�
Ý�" øZÝ
�
çIH�îUß
�
æ“fi
���
Ýhç
�
ä æ Ý�"±$
�
Ý
�
øÝ
T
çIH�îUß
�
æ“fi
���
Ýhç
T
ä æ Ý
�
$
�
Ý
T
øZÝ
U
çIH�îUß
�
æ“fi
���
Ýhç
U
ä æ Ý
T
$
�
Ý
U
øZÝ�— çIH�îUß
�
æz¥F\T¦G'&ç
*-,w*-;w"$O:036�@,/.768,K(v0d"36� H0–aFQF@903;R;M,/.F03% *: 
6‰>�03"W,/#&@Ôr!%�68,/@R*TaAq•#&@R%†J§6� FD)q•41@B,/.10$@3J§,/.768,w,/.F0
@90$;941OT,/*- 1<`%d68,9@9* aŽ.768;f6U>�03@RE;RQs0$"3*:6�O‹q•#&@9%‰\^�! o,/@R#bDF41"3*- 1<
²
ÝŽä ³´
´
´µ
�"
Ý
�
Ý
T
Ý
U
¶¸·
·
·
¹
à J
²
Œ¦º
O
ä
¦
îUß
�
³´
´
´µ
Ý�”
�
�
Ý�—
¶¸·
·
·
¹
æz¥F\T¦=?oç
6� 1D
»
ä
¦
îUß
�
³
´
´
´
µ
$
�
¦
¦ $
�
¦
¦ $
�
¦
¦ $
�
¶
·
·
·
¹
æz¥F\T¦=Doç
(v0="36� @90$(+@R*-,90‡�j2ë\V¥F\T¦G'U6�;
fi
���
²
Ýeä
»,²
Ý[ø¼J
²
Œ)º
O
æz¥F\
���
ç
tu.1*:;0–aF6�%�QFO:0U*-O:O:4F;R,/@968,/0$;c6Ž%�68,9@9*Ta†D1*Tgh0$@90$ 1"30[#&QV03@968,/#�@3\½�X6�"ƒ.
O-*: 10=#�qv6Ž%d6l,/@9* a‰D1*-qSr
q•03@R03 1"$0Z#&QV03@968,/#&@X*-;vY768;903Dd#& e6=QV#&*- 5,PD1*Tgh0$@90$ 1"30f#&Qs0$@/68,9#&@3JNY14F,P,/.F0cQV#&*: o,v#&Qs0$@/68,9#&@9;P41;R03D
q•@9#�% O:*- 10Ł,/#KO-*: 10+6�@R0u 1#�,P 10$"303;R;/6�@R*:OTEU,/.10+;96�%�08\j¤7#&@j0WaF6�%'Q1O-0�JoYV#&41 1D768@RE["$#& 1D1*T,/*:#� 1;j%d6CE
D1*-"$,ƒ6l,/0Z,/.168,Ł,/.F0wO-*: 10$;u68,u#�@u 10C6�@Ł,9.10KYs#8,9,/#�% #&@Ł,9#&Q#�q˜,/.10K%�68,/@R*Ta�YV0%�#ND1*-n103D‹\^�! ,/.F0
0WaN,9@903%'0M"36�;90=#�qL,9.10M%�68,9@9*TaŽD1*-gs03@R03 1"$0M#&QV03@968,/#�@‘@R03Q1@R03;R03 o,/*- 1<e6�;RQs0$"$,9@/6�O§%�0W,/.1#ND‹J7 1#� 10
#�qX,/.F0[O:*: F03;Z*-;c,/.10[;96�%'0�\wtu.10`%d68,9@9* a�#�Qs0$@/68,9#&@9;Z@R03Q1@R03;R03 o,/*- 1<‰,9.10U,/.F@9030–r!QV#&*: o,"$03 o,/@96�OTr
D1*Tgh0$@903 F"30�6�Q1Q1@R#?ab*-%d68,9*:#& F;Zq•#&@w6‰n7@R;R,M6� 1DH;R03"$#& 1DflD10$@9*T>�6l,/*->�0[(+*-,/.
ãZ*:@R*:"ƒ.1O-0$,KYV#&41 1D16�@RE
"3#� 1D1*-,9*:#& F;u#& ‰6[q•#�41@Br£Qs#&*- o,u%�0$;9.†6�@90
ÎNÐ ÎNЫÑmÒÔÓfÒ3ËmÌX՟ÖZÒÔÑ^ÑLÌP͑ÌXÓUÇ^Ì ×uÊXÌX͑ÉcË_×uÍfÚ �
'
fi
�
ä
¦
�
îUß
³´
´
´µ
�
¦
$w¦
�
¦
$w¦
�
¦
$w¦
�
¶ ·
·
·
¹
à�fi
���
ä
¦
îUß
�
³´
´
´µ
$
�
¦
¦ $
�
¦
¦ $
�
¦
¦ $
�
¶ ·
·
·
¹
æz¥F\
�
¦Iç
ic;†6Åq•41@R,9.103@‰0WaF6�%'Q1O:08Jc@R03Q1O:6�"30k,/.10kq•#&41@B,/.ðO:*-10A*: <�j2ë\f¥b\-¦?“¬YoE ,/.10kq•#&O-O:#?(+*: 1<ÅQV#&*- 5,
#&QV03@968,/#&@Łq•#�@+6'ò‘0341%�6� 1 ‰YV#&41 1D768@REe"$#& 1D1*T,/*:#� )柔b0$0M”N03"$,9*:#& †¥F\ “F\ çWÛ
æˆfi
���
Ýsç
U
ä
�
¥
¦
îUß
8
Ý
ß 9
—
$
�
¥
¦
î[ß
�
æ•Ý
U
$HÝ
T
ç æz¥F\
�&�
ç
(+.10$@90K,/.10KYV#&41 1D768@REe"$#& 1D1*T,/*:#� e*:;
8
Ý
ßh9
�¦¾�¿
ä
8
Ý
߁9
—
æz¥F\
�
¥oç
tu.10$ º,/.10†%d68,9@9* aÆ#&Qs0$@/68,9#&@`q•#&@d6),/.1@R030Wr£Qs#�*: o,d"$03 o,/@96�OTr£D1*-gs03@R03 1"$*: 1<
;9"ƒ.10$%�0�68,d*- o,/03@R*:#&@
QV#&*: o,/;'6� FDókž?}ƒƒ‡l„Vˆ�~9‡l€RˆN}W€WœoœV€/‡.‰&y|{'�xšyŸ‡l„Aq•#&@'6)ò‘0341%�6� 1 Æ"3#� 1D1*-,9*:#& ¬#& ¬,/.F0‰@R*:<&.o,`*-;
<&*T>&03 ‰YoE
fi
���
ä
¦
î[ß
�
³´
´
´
µ
$
�
¦
¦ $
�
¦
¦ $
�
¦
�
H�¥ $
�
H�¥
¶¸·
·
·
¹
æz¥F\
�
—5ç
�X6�"ƒ.H#�qP,/.F03;90'%�68,/@R*TaADF*-gs03@90$ 1"30`#&QV03@968,/#�@9;K*:;K6e;9254768@90'%d68,9@9* a�(+*-,9.k03O:0$%�0$ o,/;,/.76l,
6�@R0�6�O-O‘Œ$03@R#&;Ž0Wab"$03QF,Žq•#�@�,/.F#&;90A68O:#& 1<)Y76� FD1;d(+.1*-"ƒ.ª6�@90†"3O:4F;R,/0$@90$Dª6�@9#&4F 1D«,/.F0�"30$ o,/@/68O
D1*:6�<&#& 768OŸ\m™
0K"C6�O-O‹;94F"ƒ.�6[%d6l,/@9* ae6˜Àƒ�„VˆN}9ˆ'{'�xš€–yŁ‰M6� FD‰*- o,/@9#ND141"$0,/.10K 1#�,/68,/*-#& 
Áe椑 Û‹ëàÌIàú3çjä
³´
´
´
´
´
´
´
µ
Œ º
‹ ŒÄº
‹ Œ
\
\
\
º
‹ Œ
¶¸·
·
·
·
·
·
·
¹
¦
\
\
\
‘
æz¥F\
�
�&ç
(+.10$@9041;R0Z#�q&‘ *: �,/.10c6�@9<&4F%�0$ 5,_*:;v#&QF,/*-#& 76�OV6� FDŽ,/.10f*:O:O-41;R,9@/68,9*:#& `*-;_<&*T>&03 Žq•#�@_6M;R*:%'Q1O:0
xš€Wyšˆ�yš?…N‡l„V� W%�68,/@R*TaK68O-,/.F#&41<&.=6� oEK 541%UYV03@§#�q1Y768 1D1;§*:;§6fQV#&;9;R*:Y1*-O:*T,!E&\›iÅ,/@R*:D1*:6�<&#& 768O�%�68,/@R*Ta
(+*-,9.1#&4F,Ł"3#& 1;B,ƒ6� o,9;f68O:#& 1<=,/.10ZY16� 1D1;+"C68 eYs00–aFQF@903;R;90$D�6�;�ÁŽæ
²
‹ëà
²
Œ?à
²
º3ç–\jtu.10K6�@R<&41%'03 o,/;_q•#�@
6dY768 1D103DA%�68,9@9*Ta6�@90[68O-(_6?EN;f#bD1D�*: � 541%UYV03@c6� 1D�,9.10U"$03 o,/@96�OL#& F0e� éèP�?ž‘@90$q•0$@9;c,9#d,/.F0
"30$ o,/@/68O›D1*S6�<�#& 76�Oz\
�
?
Ç^ȑÉZÊcËmÌXÍÏÎ5ЫÑLÒÔӑÒ3ËmÌPÕzÖZÒÔÑ^Ñ^ÌX͑ÌPÓMÇ^Ì«ÉZÊXÊX͏×jØZÒBنÉc˧ÒW×uÓZÚ
™H0K"36� † F#I( <&03 F03@/68O:*:Œ$0#&41@+Q1@R0$>N*:#&4F;‘0–aF6�%�QFO:03;$\XãZ0$n7 F*: 1<
²
�68;
�
²
ݍä
³´
´
´
´
´
´
´µ
�"
Ý
�
Ý
T
\
\
\
ݕÅ
¶ ·
·
·
·
·
·
·
¹
æz¥F\
�
“oç
(v0="36� ‰6�Q1Q1@R#?ab*-%d68,90Z,/.10K;R03"3#� 1D�D10$@9*T>�6l,/*->�0#�q
²
�YoE
fi
���
²
Ýeä
¦
îUß
�
ÁeæÔ¦&àN$
�
àC¦Iç
²
ÝMø¼J
²
Œ)º
O
æz¥F\
�
'&ç
(+.10$@90~J
²
Œ)º
O
;B,ƒ6� FD1;Mq•#&@w,/.10d>�03"W,/#&@[.1#�O:D1*- 1<‰,9.10eãZ*-@9*:"ƒ.FO:0$,=YV#&41 1D768@RE
"3#& 1DF*-,/*-#& 1;w#& fl,/.F0
O:0Wq|,‘6� 1D@9*-<&.o,+;9*-D103;$Û
J
²
Œ)º
O
ä
¦
îUß
�
ùúÝ�”Ià
�
à5;=;=;$à
�
à/Ý�—Ÿû�Æ æz¥F\
�
?oç
�£q^(v0MQ1@R03;R"3@9*-Ys0Mòf0$41%d68 1 �YV#&41 1D16�@RE‰"3#� 1D1*-,9*:#& F;‘#� †,/.F0=@R*:<&.o,f;9*:DF0�JV6�;c*- f�j25;3\s¥F\
�
—Ž68 1D
¥F\
�&�
JF(v0Kn7 1D
fi
���
²
Ý ä
¦
îUß
�
Áeæ
²
‹7à
²
Œlà3¦Iç
²
ÝMøÇJ
²
Œ)º
O
æz¥F\
�
Doç
(+.10$@90
²
‹ ä ù ¦&à3¦&àN;=;=;Wà
�
H�¥8û
Æ
²
Œ ä ùŁ$
�
àN$
�
àN$
�
àN;=;=;$àN$
�
H�¥8û
Æ
J
²
Œ¦º
O
ä
¦
î[ß
�ÉÈ
ݕ”Ià
�
à
�
àN;=;=;–à
�
îUß
¥
8
Ý
ß
9
—ˆÊ
Æ
òf#�,9*:"$0‘,9.768,P,/.10f%d68,9@9* a[#&QV03@968,/#&@R;v<�*->&0$ ŽYoEË�X25;$\F¥F\
�
'w6� 1DŽ¥b\
�
Dw"C6�@R@RE`%'#&@90‘*: bq•#&@9%�6lr
,/*-#& e,/.768 ‰,9.10KQs#&*- o,+#&Qs0$@/68,9#&@u<&*T>&03 ‰YoE,�j2ë\s¥b\-¦G�N\P�! ˜�j25;3\V¥F\
�
'[6� 1D†¥F\
�
DFJF,/.F0=YV#&41 1D16�@RE
"3#� 1D1*-,9*:#& F;f.76C>�0UYs0$03 A41 F*:254103OTE‰;9QV03"3*Tn703Dk68 1D�*-,f*:;f"3O-0C6�@+,/.76l,c,/.F0U;/6�%'0wQs#&*- o,Z#&QV03@968,/#&@
.76�;‘YV0303 �6�Q1Q1O-*:0$D†6l,f0W>&0$@RE‰Qs#�*: o,+*: ,/.10n70$O:D‰0Wab"30$QF,6l,‘,9.10wYs#&4F 1D76�@R*:03;$\vtu.F0M6�YF*:O:*T,!E�,9#
;9QV03"$*-q|EM*: M,9.10u%d6l,/@9* a=D103@R*->868,/*T>&0Ł#�Qs0$@/68,9#&@L,/.F0u0WaF6�"$,X 168,/41@R0u#�q7,/.F0+6�Q1Q1@R#?ab*-%d68,9*:#& M68,j,/.F0
>86�@9*-#&41;fQs#&*- o,/;c*- �,/.F0Mn703O-D�*: F"3O:4FD1*: 1<',/.10MYs#&4F 1D76�@R*:03;cQs0$@9%'*-,/;+,/.F0U41;90[#8qX2N4F*-,/0M<&03 10$@/6�O
"3#� 1;R,9@941"W,/*:#� 1;u(+.1*:"ƒ.(+*:O-O‹Ys04F;90$q•4FO§O:68,/0$@u*: e"$#& 1;9*-D103@968,/*-#& 1;+#�qm;R,/6�Y1*:O-*-,!E�\
ÌIÍ
«¦§“›F§¤¨[­�§ÏÎ
Ð
ž�©k­uÑŁÒG¬[ª†§“ž@«¦¬Ó«¦Ñv§“ž@Ôk›�«¦¬[ŸÕ¯q©ˆž@¬[ª£›:›I­.ªb¨Ï›£Ÿ@›£Ôɛ£¬5§Éª†«¦¢¤¢¤›I©“®Z«.¬[ G©ff§¤«3«¦¬G›�©ˆ®L›Iª£ž¸ÖLª:©“®[­�§¤ž@­¦Ÿ
Ÿ@«NªI­�§¤žÕ«.¬Â°
ÎNÐ ÎNЫÑmÒÔÓfÒ3ËmÌX՟ÖZÒÔÑ^ÑLÌP͑ÌXÓUÇ^Ì ×uÊXÌX͑ÉcË_×uÍfÚ �
D
”b*: F"30v(v0_%�6�÷80v"$#& 1;9*-D103@96�Y1O-0v41;R0Ł#�qFYs#�,9.M%�68,/@R*Taw68 1DUQV#&*: o,§#&QV03@/6l,/#&@R;3J�*T,§*:;§*:%'Qs#&@B,ƒ6� o,
,/#'03;B,ƒ6�YFO:*:;R.‰6'@R03OS6l,/*:#� eYs0W,!(_0$03 �,9.103%‰\vi QV#&*: o,‘#&QV03@968,/#&@u*-;+<&03 10$@/6�O-O-E�(+@9*-,R,/0$ ‰q•#&@u;R#&%'0
D10$@9*->868,9*->�0c68,v,/.10f@90Wq•03@90$ 1"30QV#&*- 5,LêU*: �,/03@R%�;P#�q˜ 10$*:<&.5YV#&@9*- 1<K>�68O:410$;_#�q‹,9.10cq•41 F"$,/*-#& ‹\j¤7#&@
0WaF6�%'Q1O-0
æ“fi
�
Ýhçšíuä׋
�
ÝNí)P
�
ø~‹�"BÝbí�P�"hø]ŒƒÝNíjø]ºN"BÝNí9ü#" æz¥F\ ¥
�
ç
%'*:<&.o,XYV0+,9.10‘Qs#&*- o,v#�Qs0$@/68,9#&@^q•#&@P6Kn1@9;R,vD10$@9*->868,9*->�0�\Ltu.F0c"3#�@9@90$;9QV#& 1D1*- 1<w%d68,9@9* a[#&QV03@968,/#&@
.76�;Łq•#�@u*-,/;Ł6�@9<&4F%�0$ 5,9;u,/.10"$#b0W¨�"$*:0$ 5,9;+<&*->N*: F<U,/.10Z(v03*-<&.o,/;u,/#[,/.F0Z>�68O:410$;u#�qm,9.10Zq•41 1"W,/*-#& 
68,w,/.10'>86�@9*-#&41;wO:#N"C68,9*:#& 1;$\ei ê�r£;9.1*Tq|,M*- ),9.10dQV#&*: o,=#&QV03@/6l,/#&@w"3#&@R@90$;9QV#& 1D1;M,9#�6‰D1*S6�<�#& 76�O
;9.F*-q|,+*: e,9.10K%d68,9@9* aŽ#&QV03@968,/#�@3\jtu.541;+,/.10K%�68,/@R*Tad032541*->86�O-03 o,+#�q%�j2V\V¥F\ ¥
�
*:;
fi
�#²
ݎäBÁe戋
�
à�‹�"Wà�Œ?à�º="ƒà
�
ç
²
Ý æz¥F\ ¥F¦Iç
òf#8,/0d,9.10e6�D1DF*-,/*-#& H#8qc6†Œ303@R#A*: H,9.10Žn1q|,9.Å03O:0$%�0$ o,M(+.1*-"ƒ.Æ%�6�÷803;U*T,U"3O-0C6�@w,/.76l,ӌ'*-;M,/.F0
"3#N0$¨'"$*:03 o,‘#�qmÝNíI\
`�ab`�ab` dfpsr�g†e:Ø3g†tك |%j�r�g†tvpR}
tu.10f6�YV#I>�0f*-O:O-41;R,9@/68,903D["C68;903;P*: [(+.1*-"ƒ.',9.10‘Ys#&4F 1D76�@BE["3#& FD1*-,9*:#& 1;X68@90unFab03D‹\j�£qs,/.10+YV#&41 1DNr
6�@BEA"3#& FD1*-,9*:#& 1;w6�@R0Uœ7}W€WyŸ‡3ˆ�yŸWJ›,9.10'q•#�@9% #�qv,/.10`%�68,/@R*Ta�#&Qs0$@/68,9#&@K"ƒ.76� 1<�03;3\e_#& 1;9*-D103@K,/.F0
03*-<&.o,Rr£Qs#&*- o,LQV03@R*:#ND1*:"P%�0$;9.`;9.F#I(+ `YV03O-#I(K\mtu.1*:;L"C6� [0$*-,/.F03@^YV0ŁQ1@R03;90$ o,/03D'#& `6cO-*: 1036�@L%'03;R.
(+*-,9.`@903QV0C6l,/03D'03 o,9@9*:0$;3J5#&@j%'#&@90_;941<&<�03;R,9*->�03O-EU#& '6K"$*:@R"341O:6�@^%'03;9.�6�;j*- �¤m*:<&41@R0u¥F\ ¥F\m™ª.F03 
,/.F0[%�0$;9.A*:;cOS6�*-D�#&4F,Z#& �,/.10[QV03@R*:%'0$,903@c#8q_6Ž"$*:@R"3O:08J‹*-,cD1#N03;9 Ú� ,K@R0C6�O-O-E‰%d6l,9,/0$@f(+.10$@90[,/.F0
 541%UYV03@R*: 1<K;R,/6�@R,9;_68;PO-#& 1<w6�;X*-,

03 1D1;IŽM6l,P,9.10+Qs#�*: o,ÚÛR41;R,vQ1@R03"30$D1*: F<=*T,/;j;R,/6�@R,9*: 1<KO-#b"368,/*-#& ‹\
;=;=; ' ? ¦
�
¥ — � “ ' ? ¦
�
;=;=;
ßeä $ $
�
$ $ $ $ $ $ $
�
 $
ê[ä
�
¦ ; ; ; ; ; ; ‘
�j*:<&.o,+QV#&*: o,9;u#& �6[O-*: 1036�@ŁQV03@9*-#bDF*:"Z%'03;R.‹\Pî[ßeä
�
sHN‘
tu.10‘%d68,9@9* aU,/.168,X@90$Q1@90$;903 o,9;ŁD1*-gs03@R03 1"$*: 1<K;R"ƒ.103%'03;Pq•#&@j;9"36�OS6�@X0$2N4168,/*-#& 1;X#& �6wQV03@R*:#ND1*:"
%'03;9.w*:;˜@R0$q•03@R@90$D=,/#f6�;§6Łœ7}W€WyŸ‡3ˆ�yŸh%�68,/@R*Tas\˜i¬,!ENQ1*-"C6�O5QV03@R*:#ND1*:"L,9@9*-D1*S6�<�#& 76�O8%d68,9@9* a#&QV03@968,/#&@
(+*-,9. 1#& 541 1*Tq•#&@9% 03 o,/@R*:03;‘*:;u<�*->&0$ eq•#&@+6�“Ir!QV#&*: o,u%'03;R.†YoE
ÁkÜoæz“`Û
²
‹Và
²
ŒIà
²
º3çjä
³´
´
´
´
´
´
´
´
µ
Œ5" º
�
‹!Ý
‹�"ތ
�
º
T
‹
�
Œ
T
º
U
‹
T
Œ
U
º
V
‹
U
Œ
V