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### Lectures 8, 9 and 10

Finite Difference Discretization of Hyperbolic Equations:Linear Problems

First Order Wave Equation

INITION BOUNDARY VALUE PROBLEM (IBVP)

Initial Condition:

Boundary Conditions:

Finite DifferenceSolution

NOTATION:

- approximation to
- vector of approximate values at time ;
- vector of exact values at time ;

Consistency

The difference scheme ,

is consistent with the differential equation

if:

For all smooth functions

when .

Stability

The difference scheme is stable if:

There exists such that

for all ; and n, such that

Above condition can be written as

Lax EquivalenceTheorem

A consistent finite difference scheme for a partial differential equation for which the initial value problem is well-posed is convergent if and only if it is stable.

CFL Condition

Mathematical Domain of Dependence of

Set of points in where the initial or boundary

data may have some effect on .

Numerical Domain of Dependence of

Set of points in where the initial or boundary

data may have some effect on .

CFL Condition

CFL Condition

For each the mathematical domain of de-

pendence is contained in the numerical domain of dependence.

CFL Theorem

The CFL condition is a necessary condition for the convergence of a numerical approximation of a partial differential equation, linear or nonlinear.

Fourier Analysis

- Provides a systematic method for determining stability → von Neumann Stability Analysis
- Provides insight into discretization errors

Fourier Analysis

Fourier Modes and Properties…

Fourier mode: ( integer )

- Periodic ( period = 1 )
- Orthogonality
- Eigenfunction of

Fourier Analysis

…Fourier Modes and Properties

- Form a basis for periodic functions in
- Parseval’s theorem

Fourier Analysis

Fourier Modes and Properties…

- Basis for periodic (discrete) functions
- Parseval’s theorem

von Neumann Stability Criterion

Fourier Analysis…First Order Upwind Scheme…

amplification factor

Stability if which implies

von Neumann Stability Criterion

Fourier Analysis…FTCS Scheme

amplification factor

Unconditionally Unstable Not Convergent

Method of Lines

Generally applicable to time evolution PDE’s

- Spatial discretization

Semi-discrete scheme (system of coupled ODE’s

- Time discretization (using ODE techniques)

Discrete Scheme

By studying semi-discrete scheme we can better

understand spatial and temporal discretization errors

Method of Lines

Fourier Analysis …

Write semi-discrete approximation as

inserting into semi-discrete equation

Method of Lines

Predictor/Corrector Algorithm …

Model ODE

Predictor

Corrector

Combining the two steps you have

Method of Lines

…Predictor/Corrector Algorithm

Semi-discrete equation

Predictor

Corrector

Combining the two steps you have

Method of Lines

…Path B

- Give the same discrete Fourier equation
- Simpler
- “Decouples” spatial and temporal discretization

For each θ, the discrete Fourier equation is the

result of discretizing the scalar semi-discrete

ODE for the θ Fourier mode

Method of Lines

Absolute Stability Diagrams …

Given and complex-valued

(EF) or (EB) or …;

is defined such that

Method of Lines

…Application to the wave equation…

EF is unconditionally unstable

EB is unconditionally stable

CN is unconditionally stable

Method of Lines

…Application to the wave equation…

Stable schemes can be obtained by:

1) Selecting explicit time stepping algorithm which

have some stability on imaginary axis

2) Modifying the original equation by adding “artificial

viscosity”

Method of Lines

…Application to the wave equation…

Adding Artificial Viscosity

Additional Term

EF Time First Order Upwind

EF Time Lax-Wendroff

Method of Lines

…Application to the wave equation…

Adding Artificial Viscosity

For each Fourier mode θ,

Additional Term

Dissipation andDispersion

represents Decay

dissipation relation

represents Propagation

dispersion relation

For exact solution of

no dissipation

(constant)no dispersion

Dissipation andDispersion

- For the upwind scheme dissipation dominates over dispersion Smooth solutions
- For Lax-Wendroff and Beam-Warming dispersion is the leading error effect Oscillatory solutions ( if not well resolved)
- Lax-Wendroff has a negativephase error
- Beaming-Warming has (for ) a positive phase error

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