CFD Lecture 07 Numerical Properties of Discretized Equation(2015)

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Numerical Properties of Discretized Equation 1 
Numerical Properties of Discretized Equation 
 
The following properties are crucial to the performance of a discretization 
equation: 
 Truncation Error 
 Consistency 
 Stability 
 Conservation 
 Boundedness 
 Realizibility 
 Convergence 
Numerical Properties of Discretized Equation 2 
[Truncation Error] 
 
Consider the diffusion equation 
 
 
2
2
f f
t x
α∂ ∂=
∂ ∂
 (1) 
 
 
Using a forward difference for the time derivative and a central difference for the 
second spatial derivative (FTCS), we have the discretized algebraic equation 
(DAE) of 
 
 
1
-1 1
2
2n n n n ni i i i if f f f f
t x
α
+
+− − +=
∆ ∆
(FTCS scheme) (2) 
 
Numerical Properties of Discretized Equation 3 
By Taylor’s expansion about the grid point i 
 
 
2 2 3 3 4 4
5
1 2 3 4
( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )
2 3! 4!
n n
i i i i i i
f x f x f x ff f x O x
x x x x+
∂ ∆ ∂ ∆ ∂ ∆ ∂
= + ∆ + + + + ∆
∂ ∂ ∂ ∂
 (3) 
 
 
2 2 3 3 4 4
5
1 2 3 4
( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )
2 3! 4!
n n
i i i i i i
f x f x f x ff f x O x
x x x x−
∂ ∆ ∂ ∆ ∂ ∆ ∂
= − ∆ + − + + ∆
∂ ∂ ∂ ∂
 (4) 
 
2 2 3 3
1 4
2 3
( ) ( )( ) ( ) ( ) ( )
2! 3!
n n
i i i i i
f t f t ff f t O t
x t t
+ ∂ ∆ ∂ ∆ ∂= + ∆ + + + ∆
∂ ∂ ∂
 (5) 
 
Numerical Properties of Discretized Equation 4 
Substituting into Eq. (2) yields 
 
 
2 2 4 4
3 2 5
2 2 2 4
( )( ) ( ) ( ) [( ) ( ) ( ) ( )]
2 ( ) 4!i i i i
f t f f x fO t x O x
t t x x x
α∂ ∆ ∂ ∂ ∆ ∂
+ + ∆ = ∆ + + ∆
∂ ∂ ∆ ∂ ∂
 
 
⇒ 
2 2 2 4
3 3
2 2 4( ) ( ) ( ) ( ) ( ) ( ) 02 12
 , [ . ]
i i i i
f f t f x f O t O x
t x
PDE Truncation Error T
x
E
t
α α∂ ∂ ∆ ∂ ∆ ∂− + − + ∆ + ∆ =
∂ ∂ ∂ ∂
 
 (6) 
 
 (Expanded Differential Equation (EDE) of Discretized Algebraic Equation) 
 
⇒ 2[ . [[ ] , [ ]]] .T E O tPD xEEDE = = ∆ ∆− 
 
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Numerical Properties of Discretized Equation 5 
Further, consider Crank-Nicolson scheme for diffusion equation 
 
 
1
1 1 1
1 1 -1 -12 [ 2( ) ]2
n n
n n n n n ni i
i i i i i i
f f f f f f f f
t x
α+ + + +
+ +
−
= + − + + +
∆ ∆
 (7) 
 
The expanded differential equation of Eq. (7) can be obtained as 
 
 
2 2 3
2 2
2 2 2( ) ( ) ( ) ( ) [ , ]2 2i i i i
f f t f t f O t x
t x t t x
α α∂ ∂ ∆ ∂ ∆ ∂− = − + + ∆ ∆
∂ ∂ ∂ ∂ ∂
 (8) 
 
⇒ 2[ . .] [ , ]T E O t x= ∆ ∆ 
 
However, we have 
 
 
2 3 2
2 2 2( ) ( ) [( ) ( ) ]2 2 2i i i i
t f t f t f f
t t x t t x
α α∆ ∂ ∆ ∂ ∆ ∂ ∂ ∂− + = − −
∂ ∂ ∂ ∂ ∂ ∂
 (9) 
Numerical Properties of Discretized Equation 6 
Thus, Eq. (8) takes the form 
 
 
2 2
2 2
2
0
( ) ( ) [( ) ( ) ] [( ) ] [( ) ]
2i i i i
f f t f f O t O x
t x t t x
α α
=
∂ ∂ ∆ ∂ ∂ ∂
− + − = ∆ + ∆
∂ ∂ ∂ ∂ ∂

 (10) 
 
It follows that the truncation error for the Crank-Nicolson scheme is 
 
2 2[ . .] [ , ]T E O t x= ∆ ∆ 
 
 The leading terms in the truncation error should be examined very carefully to 
see if they can be identified as a multiple of a derivation of the original PDE. If 
they can, they should be replaced by expressions of higher order! 
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Numerical Properties of Discretized Equation 7 
[Consistency] 
 
A descritized algebraic equation (DAE) analogous to a PDE is said to be 
“consistent” if 
 
 
0
0
lim [ . .] 0
x
t
T E
∆ →
∆ →
= 
 
For example, the Dufort-Frankel (1953) finite difference scheme for the diffusion 
equation takes the form 
 
 
1 1
1 1
1 -12 [ ( ) ]2
n n
n n n ni i
i i i i
f f f f f f
t x
α+ − + −
+
−
= − + +
∆ ∆
 (11) 
 
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Numerical Properties of Discretized Equation 8 
The leading terms in the truncation error for Eq. (11) are 
 
 

4 2 3
2 2 2
4 2 3
2
1[ . .] ( ) ( ) ( ) ( ) ( ) ( )
12 6i i i
f f t fT E x t
x t x t
α α
β
∂ ∂ ∆ ∂
= ∆ − − ∆ +
∂ ∂ ∆ ∂
 (12) 
 
 
2
2
20
0
lim [ . (12)] ( ) 0ix
t
fEq
t
αβ
∆ →
∆ →
∂
= − ≠
∂
 (13) 
 
where ( / )t x β∆ ∆ = . 
 
It follows that the Dufort-Frankel scheme is consistent with a hyperbolic equation 
 
 
2 2
2 2
f f f
t t x
αβ α∂ ∂ ∂+ =
∂ ∂ ∂
 (14)
Numerical Properties of Discretized Equation 9 
[Stability] (Marching Stability) 
 
During numerical simulation for a marching PDE, errors may be accumulated as 
a result of 
 Round-off 
 Truncation 
 Mistakes 
 
 A DAE is said to be “stable” if the cumulative errors are not permitted to 
grow in the sequence of numerical procedures as the calculation proceeds from 
one marching step to the next. 
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Numerical Properties of Discretized Equation 10 
In practice, the stability of a numerical solution may imply the following 
properties of numerical solution: 
 
 Conservation 
 Boundedness 
 Realizibility 
 
Numerical Properties of Discretized Equation 11 
[Conservation] – A discretization scheme possesses the conservation property if 
it preserves certain integral conservation relations of the continuum equation. 
 
Conservation laws of physical quantities, such as mass, momentum, energy, 
species, …, at the discrete level are satisfied adequately, which can only be 
distributed improperly. 
 
 
V S V convection diffusion
accumulation influx generation
dV f ndS dV f
t
φ φ φ α φ∂ = − ⋅ + = − ∇
∂ ∫ ∫ ∫ , where v
 
 

  
 
 
 Conservative discretization does generally give more accurate solution. 
 
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Numerical Properties of Discretized Equation 12 
 Non-conservative discretization may produce reasonably looking results which 
are totally wrong 
 
 even non-conservative schemes can be consistent and stable 
 correct solutions are recovered in the limits of very fine grids 
 
 Any finite volume discretization is conservative by construction both locally 
(over every single control volume) and globally (for the whole domain) 
 
 The conservation property of a finite difference discretization depends on 
 
 the form of continuum equation used 
 the finite difference scheme 
 
Numerical Properties of Discretized Equation 13 
 A finite difference discretization is conservative if it can be written in the form 
 
1
1/ 2 1/ 2
n n
i i i i
i
f f
t x
φ φ
φ
+
+ −− −+ =
∆ ∆
 i = I1….., Im 
 
Summation of the finite difference over the grid points gives 
 
 
1
1
1/ 2 1/ 2[ ]
mI n n
i i i i
i
i I
f f x
t x
φ φ
φ
+
+ −
=
− −
+ = ∆
∆ ∆∑  
 
 
1 1 1
1
1/ 2 1/ 2
1 [( ) ] ( )
m m mI I I
n n
i i i i i
i I i I i I
x f f x
t
φ φ φ+ + −
= = =
− ∆ + − = ∆
∆ ∑ ∑ ∑  
 
Numerical Properties of Discretized Equation 14 
1 1 1 1 1 1
1
1
1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 3/ 2 3/ 2 5/ 2
3/ 2 1/ 2 1/ 2 1/ 2
1/ 2 1/ 2
( )
m
m m m m
m
I
i i I I I I I I
i I
I I I I
I I
f f f f f f f f
f f f f
f f
+ − − + + + + +
=
− − − +
− +
− = − + − + − + −
− + − +
= − +
∑ 
 
 
⇒ 
1
1 1
1
1/ 2 1/ 2
inf
1 [( ) ]
m m
m
I I
n n
i i I I i
i I i Ilux
generation
x f f x
t
φ φ φ+ − +
= =
− ∆ = − + ∆
∆ ∑ ∑ 

 
 
i.e. The time rate of accumulation of φ over the solution domain equals the net 
influx of φ through the control surface plus total generation of φ within the domain. 
Numerical Properties of Discretized Equation 15 
[Boundedness] - Numericalsolutions should lie within proper bounded 
 
 Physically non-negative variables, e.g. density, kinetic energy of turbulence, etc., must 
always be positive 
 In the absence of sources, some equations require that the minimum and maximum 
values of the variable be found on the boundaries of the domain (i.e., elliptic PDE, e.g., 
the heat equation for the temperature when no heat sources are present) 
 Boundedness is difficult to guarantee 
 Only some first order schemes guarantee this property 
 All higher-order schemes prone to producing unbounded/overshot solutions may 
have “stability” and “convergence” problems, which may be avoided by means of 
grid refinement (at least locally) 
Numerical Properties of Discretized Equation 16 
[Realizibility] – Numerical solutions should be physically realistic. 
 
 Model of phenomena, which are too complex to treat directly (e.g., turbulence, 
combustion, or multiphase flow), should be designed to guarantee physically 
realistic solutions. 
 
 Sometimes “instability” can be identified with a physical “implausibility”, i.e. 
unacceptable results in physical sense. 
 
Numerical Properties of Discretized Equation 17 
Consider the FTCS scheme for a conduction equation 
 
 

1
i i 1 i 1 i i 1 i 1 i2 2( ) (1 2 ) ( ) (1 2 )
n n n n n n nt tT T T T T T T
x x
τ
α α τ τ+ + − + −
∆
∆ ∆
= + + − = ∆ + + − ∆
∆ ∆
 (14) 
 
where 2( / )t xτ α∆ = ∆ ∆ . 
 
Suppose at a time t = n∆t, 1 1 100 
n n o
i iT T C+ −= = and 0n oiT C= . If 1τ∆ = , 
then Eq. (14) gives 
 
 1 200 n oiT C
+ = (Physically impossible value!) 
 
Numerical Properties of Discretized Equation 18 
[Convergence] 
 
A solution to a discretized algebraic equation (DAE) which approximates a given 
PDE is said to be “convergence” if, at each grid point in the solution domain the 
numerical solution approaches the true/analytical solution of the P.D.E. having the 
same initial and boundary conditions as the grid sizes tend to zero. 
 
 
∆x→0, ∆t→0 Numerical Solution 
to DAE 
True/Analytical Solution 
to PDE 
 
Numerical Properties of Discretized Equation 19 
[Lax’s Equivalence Theorem] 
 
Given a properly posed initial value problem and a finite-difference 
approximation to it that satisfies the consistency condition, stability is the 
necessary and sufficient condition for convergence. 
 
[Consistency] + [Stability] ⇒ [Convergence] 
 
 Most “real” flow problems are nonlinear and are boundary or mixed 
initial/boundary value problems so that the Lax equivalence theorem cannot 
always be applied rigorously. Consequently the Lax equivalence theorem 
should be interpreted as providing necessary, but not always sufficient, 
conditions. 
Numerical Properties of Discretized Equation 20 
 
 
Governing 
PDE 
Discretization 
System 
of DAEs 
Consistency 
Stability 
Numerical 
Solution 
True 
Solution 
Convergence 
as ∆x, ∆t → 0 
 
 
[Consistency] + [Stability] ⇒ [Convergence] 
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Numerical Properties of Discretized Equation 21 
[Accuracy] 
 
Accuracy of numerical solution to a PDE is closely related with three kinds of 
errors: 
 
(1) Modeling error - the difference between the actual physical phenomenon 
and the analytical/true solution of the mathematical model. 
(2) Discretization error - the difference between the analytical/true solution of 
the PDE and the exact solution of the system of DAEs, due to truncation 
error and errors introduced by the treatment of boundary conditions. 
(3) Convergence error - the difference between the “iterative” and “exact” 
solutions of the system of DAEs, due to run-off errors. 
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要盡量減少
Numerical Properties of Discretized Equation 22 
 
 
True Solution 
to PDE 
Numerical Solution 
to DAE 
∼ Discretization Error 
Convergence 
Error 
 
 
Numerical Properties of Discretized Equation 23 
 
 
 
 
END 
 
The path towards good numerical solutions is 
paved with uncertainty, trial and error.