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Lectures 8, 9 and 10

Lectures 8, 9 and 10. Finite Difference Discretization of Hyperbolic Equations: Linear Problems. First Order Wave Equation. INITION BOUNDARY VALUE PROBLEM (IBVP) Initial Condition: Boundary Conditions:. Solution. First Order Wave Equation. Characteristics. General solution.

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Lectures 8, 9 and 10

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  1. Lectures 8, 9 and 10 Finite Difference Discretization of Hyperbolic Equations:Linear Problems

  2. First Order Wave Equation INITION BOUNDARY VALUE PROBLEM (IBVP) Initial Condition: Boundary Conditions:

  3. Solution First Order Wave Equation Characteristics General solution

  4. Solution First Order Wave Equation

  5. Solution First Order Wave Equation

  6. Stability First Order Wave Equation

  7. Model Problem Initial condition: Periodic Boundary conditions: constant

  8. Example Model Problem Periodic Solution (U>0)

  9. Discretization Finite DifferenceSolution Discretize (0,1) into J equal intervals And (0,T) into N equal intervals

  10. Discretization Finite DifferenceSolution

  11. Discretization Finite DifferenceSolution NOTATION: • approximation to • vector of approximate values at time ; • vector of exact values at time ;

  12. Approximation Finite DifferenceSolution For example … for ( U > 0 ) Forward in Time Backward (Upwind) in Space

  13. First Order Upwind Scheme Finite DifferenceSolution suggests … Courant number C =

  14. First Order Upwind Scheme Finite DifferenceSolution Interpretation Use Linear Interpolation j – 1, j

  15. First Order Upwind Scheme Finite DifferenceSolution Explicit Solution no matrix inversion exists and is unique

  16. First Order Upwind Scheme Finite DifferenceSolution Matrix Form We can write

  17. First Order Upwind Scheme Finite DifferenceSolution Example

  18. Definition Convergence The finite difference algorithm converges if For any initial condition .

  19. Definition Consistency The difference scheme , is consistent with the differential equation if: For all smooth functions when .

  20. First Order Upwind Scheme Consistency Difference operator Differential operator

  21. First Order Upwind Scheme Consistency First order accurate in space and time

  22. Truncation Error Insert exact solution into difference scheme Consistency

  23. Definition Stability The difference scheme is stable if: There exists such that for all ; and n, such that Above condition can be written as

  24. First Order Upwind Scheme Stability

  25. First Order Upwind Scheme Stability

  26. Stability Stable if Upwind scheme is stable provided

  27. Lax EquivalenceTheorem A consistent finite difference scheme for a partial differential equation for which the initial value problem is well-posed is convergent if and only if it is stable.

  28. Proof Lax EquivalenceTheorem ( first order in , )

  29. First Order Upwind Scheme Lax EquivalenceTheorem • Consistency: • Stability: for • Convergence or and are constants independent of ,

  30. First Order Upwind Scheme Lax EquivalenceTheorem Example Solutions for: (left) (right) Convergence is slow !!

  31. Domains of dependence CFL Condition Mathematical Domain of Dependence of Set of points in where the initial or boundary data may have some effect on . Numerical Domain of Dependence of Set of points in where the initial or boundary data may have some effect on .

  32. Domains of dependence CFL Condition First Order Upwind Scheme Analytical Numerical ( U > 0 )

  33. CFL Theorem CFL Condition CFL Condition For each the mathematical domain of de- pendence is contained in the numerical domain of dependence. CFL Theorem The CFL condition is a necessary condition for the convergence of a numerical approximation of a partial differential equation, linear or nonlinear.

  34. CFL Theorem CFL Condition Stable Unstable

  35. Fourier Analysis • Provides a systematic method for determining stability → von Neumann Stability Analysis • Provides insight into discretization errors

  36. Continuous Problem Fourier Analysis Fourier Modes and Properties… Fourier mode: ( integer ) • Periodic ( period = 1 ) • Orthogonality • Eigenfunction of

  37. Continuous Problem Fourier Analysis …Fourier Modes and Properties • Form a basis for periodic functions in • Parseval’s theorem

  38. Continuous Problem Fourier Analysis Wave Equation

  39. Discrete Problem Fourier Analysis Fourier Modes and Properties… Fourier mode: , k ( integer )

  40. Discrete Problem Fourier Analysis …Fourier Modes and Properties… Real part of first 4 Fourier modes

  41. Discrete Problem Fourier Analysis …Fourier Modes and Properties… • Periodic (period = J) • Orthogonality

  42. Discrete Problem Fourier Analysis …Fourier Modes and Properties… • Eigenfunctions of difference operators e.g.,

  43. Discrete Problem Fourier Analysis Fourier Modes and Properties… • Basis for periodic (discrete) functions • Parseval’s theorem

  44. von Neumann Stability Criterion Fourier Analysis Write Stability Stability for all data

  45. von Neumann Stability Criterion Fourier Analysis First Order Upwind Scheme…

  46. von Neumann Stability Criterion Fourier Analysis …First Order Upwind Scheme… amplification factor Stability if which implies

  47. von Neumann Stability Criterion Fourier Analysis …First Order Upwind Scheme Stability if:

  48. von Neumann Stability Criterion Fourier Analysis FTCS Scheme… Fourier Decomposition:

  49. von Neumann Stability Criterion Fourier Analysis …FTCS Scheme amplification factor Unconditionally Unstable Not Convergent

  50. Time Discretization Lax-WendroffScheme Write a Taylor series expansion in time about But …

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