Download - Consider the initial boundary value problem
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.