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Consider the initial boundary value problem

Consider the initial boundary value problem. Weak Formulation. Semidiscrete problem. Matrix Form. Weak Formulation. Matrix Form. Full Discritization. Von Neumann method for stability Fourier Series method. How to Examine the stability of a FD scheme:. Subsitute.

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Consider the initial boundary value problem

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  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 12equationsin12unknowns

  5. Some Other Classes of Numerical Methods Least Square Method consider Is minimized 12equationsin12unknowns

  6. First Order Hyperbolic Equation (FDM) consider FD for the above equation CD on x Unstable a>0 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 Discrete Problem PDE Properties For Continuous Problem PDE inherited Examples: Maximum principle for elliptic Domain of dependence

  9. 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)

  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. (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

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