Consider the initial boundary value problem

Weak Formulation

Matrix Form

Semidiscrete problem

Weak Formulation

Matrix Form

Full Discritization

How to Examine the stability of a FD scheme:

Von Neumann method for stabilityFourier Series method

Subsitute

Then necessary and sufficient condition for stability

Example: (#3 in HW4)

Consider

Show that it is always unstable (for all r)

Identities

Collocation Method

Some Other Classes of Numerical Methods

consider

12 equations in 12 unknowns

Least Square Method

Some Other Classes of Numerical Methods

consider

Is minimized

12 equations in 12 unknowns

First Order Hyperbolic Equation (FDM)

consider

FD for the above equation

CD on x

BWD on x

FWD on x

Unstable a>0

Most well known formula is the Lax-Wedroff formula

Hyperbolic Equation (FDM)

consider

For stability

interperetation

Numerical Method

PropertiesFor

Continuous ProblemPDE

PropertiesFor

Discrete ProblemPDE

inherited

Maximum principle for elliptic

Domain of dependence

Examples:

Hyperbolic Equation (FDM)

consider

interperetation

Use Fourier transform to solve the problem and we get

This solution depends on values of f(x) at the endpoints and on values of g(x) on the interval

Slope = 1 Slope = -1

The region of determination of the solution at (xm,tn)

Initial data outside the region does not affect the solution at (xm,tn)

Hyperbolic Equation (FDM) interperetation

|Slope| = 1

Theorem: The numerical solution at (xm,tn) cannot in general converge to the exact solution at (xm,tn) unless the numerical interval of dependence includes the analytic interval of dependence.

CFL condition Courant-Friedrichs-Lewy

(Stability condition is violated) initial data needed is not all used and available which means convergence is not possible

(Stability condition is satisfied) all needed initial condition to find the analytic solution are available.