lectures 8, 9 and 10

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:


Solution

Characteristics

General solution


Solution


Stability

Model Problem

Initial condition:

Periodic Boundary conditions:

constant

Model ProblemExample

Periodic Solution (U>0)

Finite DifferenceSolution

Discretization

Discretize (0,1) into J equal intervals

And (0,T) into N equal intervals


Discretization


Discretization

NOTATION:• approximation to

• vector of approximate values at time ;

• vector of exact values at time ;


Approximation

For example … for ( U > 0 )

Forward in Time Backward (Upwind) in Space


First Order Upwind Scheme

suggests …

Courant number C =



Interpretation

Use Linear Interpolation

j – 1, j



Explicit Solution

no matrix inversion

exists and is unique



Matrix Form

We can write



Example

ConvergenceDefinition

The finite difference algorithm converges if

For any initial condition .

ConsistencyDefinition

For all smooth functions

when .

The difference scheme ,

is consistent with the differential equation

if:

ConsistencyFirst Order Upwind Scheme

Difference operator

Differential operator

ConsistencyFirst Order Upwind Scheme

First order accurate in space and time

Truncation Error

Insert exact solution into difference scheme

Consistency

Stability

The difference scheme is stable if:

There exists such that

for all ; and n, such that

Definition

Above condition can be written as

StabilityFirst Order Upwind Scheme

Stability

Stable if

Upwind scheme is stable provided

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.


Proof

( first order in , )



• Consistency:• Stability: for• Convergence

or

and are constants independent of ,



Example

Solutions for:

(left)

(right)

Convergence is slow !!

CFL ConditionDomains of dependence

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 ConditionDomains of dependence


Analytical Numerical ( U > 0 )

CFL ConditionCFL Theorem

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.

CFL ConditionCFL Theorem

Stable Unstable

Fourier Analysis

• Provides a systematic method for determining stability → von Neumann Stability Analysis

• Provides insight into discretization errors

Fourier AnalysisContinuous Problem

Fourier Modes and Properties…

Fourier mode: ( integer )

• Periodic ( period = 1 )

• Orthogonality

•Eigenfunction of


…Fourier Modes and Properties

• Form a basis for periodic functions in

• Parseval’s theorem


Wave Equation

Fourier AnalysisDiscrete Problem


Fourier mode: ,

k ( integer )


…Fourier Modes and Properties…

Real part of first 4 Fourier modes



• Periodic (period = J)

• Orthogonality



• Eigenfunctions of difference operators e.g.,



• Basis for periodic (discrete) functions

• Parseval’s theorem

Fourier Analysisvon Neumann Stability Criterion

Write

Stability

Stability for all data


First Order Upwind Scheme…


…First Order Upwind Scheme…

amplification factor

Stability if which implies


…First Order Upwind Scheme

Stability if:


FTCS Scheme…

Fourier Decomposition:


amplification factor

Unconditionally Unstable Not Convergent

…FTCS Scheme

Lax-WendroffScheme

Time Discretization

Write a Taylor series expansion in time about

But …

Lax-WendroffScheme

Spatial Approximation

Approximate spatial derivatives

Lax-WendroffScheme

Equation

no matrix inversion

exists and is unique

Lax-WendroffScheme

Interpretation

Use Quadratic

Interpolation

Lax-WendroffScheme

Analysis

Consistency

Second order accurate in space and time

Lax-WendroffScheme

Analysis

Truncation Error

Consistency

Insert exact solution into difference scheme

Lax-WendroffScheme

Analysis

Stability

Stability if:

Lax-WendroffScheme

Analysis

Convergence

• Consistency:

• Stability:

• Convergence

and are constants independent of

Lax-WendroffScheme

Domains of Dependence

Analytical Numerical

Lax-WendroffScheme

CFL Condition

Stable Unstable

Lax-WendroffScheme

Example

Solutions for:

C = 0.5

= 1/50 (left)

= 1/100 (right)

Lax-WendroffScheme

Example

= 1/100

C = 0.5

Upwind (left)

vs.

Lax-Wendroff (right)

Beam-WarmingScheme

Derivation

Use Quadratic

Interpolation

Beam-WarmingScheme

Consistency and Stability

• Consistency,

• Stability

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

Notation

approximation to

vector of semi-discrete approximations;

Method of LinesSpatial Discretization

Central difference … (for example)

or, in vector form,


Write semi-discrete approximation as

inserting into semi-discrete equation

Fourier Analysis …


For each θ, we have a scalar ODE

… Fourier Analysis …

Neutrally stable


Exact solution

Semi-discrete solution

… Fourier Analysis …


Fourier Analysis …

Method of LinesTime Discretization

Predictor/Corrector Algorithm …

Model ODE

Predictor

Corrector

Combining the two steps you have

Method of LinesTime Discretization

…Predictor/Corrector Algorithm

Semi-discrete equation

Predictor

Corrector

Combining the two steps you have

Method of LinesFourier Stability Analysis

Fourier Transform


Application factor

with

Stability


PDE

Semi-discrete

Semi-discrete Fourier

Discrete

Discrete Fourier


Path B …

Semi-discrete

Fourier semi-discrete

Predictor

Corrector

Discrete


…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 LinesMethods for ODE’s

Model equation: complex- valued

Discretization

EF

EB

CN


Given and complex-valued

Absolute Stability Diagrams …

(EF) or (EB) or … ;

is defined such that


…Absolute Stability Diagrams …

EF

EB

CN


…Absolute Stability Diagrams


Application to the wave equation…

For each

Thus,

• (and ) is purely imaginary • for


…Application to the wave equation…

EF is unconditionally unstable

EB is unconditionally stable

CN is unconditionally stable



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”



Explicit Time Stepping Scheme

Predictor/Corrector



Explicit Time Stepping Scheme

4 Stage Runge-Kutta



Adding Artificial Viscosity

Additional Term

EF Time First Order Upwind

EF Time Lax-Wendroff



Adding Artificial Viscosity

For each Fourier mode θ,

Additional Term


…Application to the wave equation

Lax-Wendroff Scheme

Dissipation andDispersion

Model Problem

with and periodic boundary conditions.

Solution


Model Problem

represents Decay

dissipation relation

represents Propagation

dispersion relation

For exact solution of

no dissipation

(constant) no dispersion


Modified Equation

First Order Upwind

Lax-Wendroff

Beam-Warming


Modified Equation

• 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 negative phase error• Beaming-Warming has (for ) a positive phase

error


Examples

First Order Upwind


Examples

Lax-Wendroff (left)

vs.

Beam-Warming (right)


Exact Discrete Relations

For the exact solution

, and

For the discrete solution

, and

lectures 8, 9 and 10

Documents

thatdefinitionabove

necessary condition

theoremthe cfl condition

initial value problem

numerical approximation

partial differential

time vector of exact

periodic boundary conditions