Consider the initial boundary value problem

1
Consider the initial boundary value problem
Weak Formulation
Semidiscrete problem
Matrix Form
2
Weak Formulation
Matrix Form
Full Discritization
3
Von Neumann method for stability Fourier Series
method
How to Examine the stability of a FD scheme
Subsitute
Then necessary and sufficient condition for
stability
Example (3 in HW4)
Consider
Show that it is always unstable (for all r)
Identities
4
Some Other Classes of Numerical Methods
Collocation Method
consider
12 equations in 12 unknowns
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Some Other Classes of Numerical Methods
Least Square Method
consider
Is minimized
12 equations in 12 unknowns
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First Order Hyperbolic Equation (FDM)
consider
FD for the above equation
CD on x
Unstable agt0
BWD on x
FWD on x
Most well known formula is the Lax-Wedroff formula
7
Hyperbolic Equation (FDM)
consider
For stability
interperetation
8
Numerical Method
Properties For Continuous Problem PDE
Properties For Discrete Problem PDE
inherited
Examples
Maximum principle for elliptic
Domain of dependence
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Hyperbolic Equation (FDM)
interperetation
consider
Use Fourier transform to solve the problem and we
get
Slope 1
Slope -1
This solution depends on values of f(x) at the
endpoints and on values of g(x) on the interval
The region of determination of the solution at
(xm,tn)
Initial data outside the region does not affect
the solution at (xm,tn)
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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.
(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.
CFL condition Courant-Friedrichs-Lewy