1. Consider the initial boundary value problem Weak Formulation Matrix Form Semidiscrete problem

2. Weak Formulation Matrix Form Full Discritization

3. How to Examine the stability of a FD scheme: Von Neumann method for stability Fourier 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

4. Collocation Method Some Other Classes of Numerical Methods consider 12 equations in 12 unknowns

5. Least Square Method Some Other Classes of Numerical Methods consider Is minimized 12 equations in 12 unknowns

6. 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

7. Hyperbolic Equation (FDM) consider For stability interperetation

8. Numerical Method Properties For Continuous Problem PDE Properties For Discrete Problem PDE inherited Maximum principle for elliptic Domain of dependence Examples:

9. 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 = 1Slope = -1 The region of determination of the solution at (xm,tn) Initial data outside the region does not affect the solution at (xm,tn)

10. 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.