Review of Finite Difference Methods | MATH 593, Study notes of Mathematics

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Numerical Methods for Hyperbolic

Conservation Laws

Review of Finite Difference Methods

Dr. Aamer Haque

http://math.iit.edu/~ahaque

ahaque7@iit.edu

Illinois Institute of Technology

June 11, 2009

Outline



Finite Difference Stencils



Consistency



Stability



Convergence

Order Symbols

Order symbols are used to compare the relative behavior of

two functions and as

is (“Big O”) if

is (“Little o”) if

We will mainly use “Big O” notation

O  g  x

lim

x  a

f  x

g  x

=L≠ 0

f  x

o g  x

lim

x  a

f  x

g  x

g  x

f  x

f  x

x  a

Basic Finite Differences

f '  x

i

f  x

i 1

− f  x

i

 x

O  x

 x=x

i 1

−x

i

,∀ i

f '  x

i

f  x

i

− f  x

i− 1

 x

O  x

f '  x

i

f  x

i 1

− f  x

i− 1

2  x

O  x

2



We need approximations for 1

st

derivatives

Forward

Difference

Backward

Difference

Centered

Difference

Forward Euler

U

i

n 1

=U

i

n

 t

2  x

A



U

i 1

n

−U

i− 1

n



Unstable! (i.e. Useless)

x

i

x

i 1

x

i− 1

x

i 2

x

i− 2

t

n 1

t

n 2

t

n

 x

 t

Backward Euler

U

i

n 1

=U

i

n

 t

2  x

A



U

i 1

n 1

−U

i− 1

n 1



x

i

x

i 1

x

i− 1

x

i 2

x

i− 2

t

n 1

t

n 2

t

n

 x

 t

Implicit (i.e. Requires Matrix Solves)

Right One-Sided

U

i

n 1

=U

i

n

t

 x

A



U

i 1

n

−U

i

n



x

i

x

i 1

x

i− 1

x

i 2

x

i− 2

t

n 1

t

n 2

t

n

 x

 t

Stability depends on A

Leapfrog

U

i

n 1

=U

i

n− 1

 t

 x

A



U

i 1

n

−U

i− 1

n



x

i

x

i 1

x

i− 1

x

i 2

x

i− 2

t

n

t

n 1

t

n− 1

 x

 t

3-Level Method

Lax-Wendroff

U

i

n 1

=U

i

n

 t

2  x

A



U

i 1

n

−U

i− 1

n



 t 

2

2  x

2

A

2



U

i 1

n

− 2 U

i

n

U

i− 1

n



x

i

x

i 1

x

i− 1

x

i 2

x

i− 2

t

n 1

t

n 2

t

n

 x

 t

Beam-Warming

x

i

x

i 1

x

i− 1

x

i 2

x

i− 2

t

n 1

t

n 2

t

n

 x

 t

 t 

2

2  x

2

A

2



U

i

n

− 2 U

i− 1

n

U

i− 2

n



U

i

n 1

=U

i

n

 t

2  x

A



3 U

i

n

− 4 U

i− 1

n

U

i− 2

n



Norm Continuous Variable Discrete Points Comments

Infinity Inappropriate

Sup

1 Natural

2 Fourier

Analysis

Norms

We will only use the 1-Norm

∥u∥

1

∫

∣u x∣dx

∥u∥

2

[∫

u

2

 x dx

]

1

2

∥u∥

∞

= max

x

∣u x∣ ∥u∥

∞

= max

i

∣

u

i

∣

∥u∥

1

=h

∑

i

∣

u

i

∣

∥u∥

2

[

h

∑

i

u

i

2

]

1

2

Local Truncation Error & Consistency

L

k

 x , t=

k

[

u  x , tk − H

k

u : ,t  ; x

]

Local truncation error is obtained by plugging the actual

solution or into the finite difference method

For a 2-level method, local truncation error is defined as

The method is consistent if

lim

k  0

∥L

k

: ,t ∥= 0

L

k

n

k

[

V

n 1

− H

k

V

n

]

lim

k  0

∥L

k

n

u x ,t  V

n

Stability

An initial value problem is stable if

A finite difference method is stable if

Alternative relations are

∥u x ,t ∥≤C e

 t

∥u x , 0 ∥

∥U

n

∥≤C e

 k n

∥U

0

∥ H

k

∥≤ 1  k , ∀ k k

0

∥ H

k

n

∥≤ 1  k 

n

≤e

 k n

, ∀ kk

0

Lax Equivalence Theorem

A linear finite difference method that is stable and

accurate of order (p,q) is convergent of order (p,q)

Linear + Consistency + Stability -> Convergence!