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Finite Difference Methods for Conservation Laws: 1D Linear Differential Equation

This lecture discusses various finite difference methods for solving 1D linear differential equations in conservation laws. Topics covered include one-sided backward, one-sided forward, Lax-Wendroff, Lax-Friedrichs, and Beam-Warming difference equations. The convergence, error functions, and numerical stability of these methods are also explored. Nonlinear equations, numerical diffusion, and numerical dispersion are discussed as well.

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Finite Difference Methods for Conservation Laws: 1D Linear Differential Equation

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Presentation Transcript

  1. Lecture 2

  2. Finite Difference (FD) Methods Conservation Law: 1D Linear Differential Equation:

  3. FD Methods One-sided backward Leapfrog One-sided forward Lax-Wendroff Lax-Friedrichs Beam-Warming

  4. One-sided backward Difference equation: i+1 time stepping i i-1 j j+1 j-1

  5. One-sided forward Difference equation: i+1 time stepping i i-1 j j+1 j-1

  6. Lax-Friedrichs Difference equation:

  7. Leapfrog Difference equation:

  8. Lax-Wendroff Difference equation:

  9. Beam-Warming Difference equation:

  10. Time stepping 1st order methods in time: U(i+1) = F{ U(i) } 2nd order methods in time: U(i+1) = F{ U(i) , U(i-1) } Starting A) U(2) = F{ U(1) } ( e.g. One-Sided ) method: B) U(3) = F{ U(2) , U(1) } ( Leapfrog )

  11. Convergence - Analytical solution - Numerical solution Error function: Numerical convergence (Norm of the Error function):

  12. Norms Norm for conservation laws: Other Norms (e.g. spectral problems)

  13. Numerical Stability Courant, Friedrichs, Levy (CFL, Courant number) Upwind schemes for discontinuity: a > 0 : One sided forward a < 0 : One sided backward

  14. Lax Equivalence Theorem For a well-posed linear initial value problem, the method is convergent if and only if it is stable. Necessary and sufficient condition for consistent linear method Well posed: solution exists, continuous, unique; • numerical stability (t) • numerical convergence (xyz)

  15. Discontinuous solutions Advection equation: Initial condition: Analytic solution:

  16. Analytic vs Numerical Dx = 0.01

  17. Analytic vs Numerical Dx = 0.001

  18. Nonlinear Equations Linear conservation: Nonlinear conservation:

  19. Nonlinear FD methods Lax-Friedrichs linear stencil: nonlinear stencil:

  20. Nonlinear FD methods Lax-Wendroff MacCormack’s two step method:

  21. FD Methods Methods to supress numerical instabilities Numerical diffusion (first order methods)

  22. Numerical Dispersion Lax-Wendroff (or second order methods) Beam-Warming method

  23. Numerical dispersion ( a > 0 ) ( a < 0 )

  24. Numerical Dispersion ( a > 0 ) ( a < 0 )

  25. Convergence Lax-Friedrichs: Convergence:

  26. FD Methods + / Primitive + / Fast • / Accuracy • / Numerical instabilities

  27. end www.tevza.org/home/course/modelling2012

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