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P DE’s Discretization

P DE’s Discretization. Sauro Succi. Now to actual PDE’s. PDE’s Family. Diffusion Equation. Advection-Diffusion Equation. Advection-Diffusion-Reaction Equation. Diffusion Equation. DE: Dispersion Relation. Dispersion Relation: Fourier. Fourier Transform: plane-wave superposition.

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P DE’s Discretization

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  1. PDE’sDiscretization Sauro Succi

  2. Now to actualPDE’s

  3. PDE’s Family Diffusion Equation Advection-Diffusion Equation Advection-Diffusion-Reaction Equation

  4. Diffusion Equation

  5. DE: Dispersion Relation

  6. Dispersion Relation: Fourier Fourier Transform: plane-wave superposition

  7. DE: Dispersion Relations Continuum Discrete

  8. Dispersion Relation: general Propagation/Dispersion: Diffusion/Hyperdiffusion

  9. NumericalDiffusion: positive = overdamping

  10. Numericaldispersion: Gibbsphenomena

  11. Diffusion Equation: Forward Euler

  12. DE: centered differences

  13. Computational molecules Elementarytrimer (stencil) builds up a crystal (matrix)

  14. DE: Centered-Euler

  15. DE: Centered-Euler

  16. DE: Stability analysis

  17. DE: Stability analysis

  18. DE: Linear stability analysis

  19. DE: Discrete DR Square and Sum: Stable: 2nd order accurate

  20. Discrete DR: Dispersion Divide: Exact

  21. Entry Exercise: Heatdiffusion in 1d wire

  22. Advection Equation

  23. Dispersion Relation

  24. AE: Euler centered 1st order centered: unconditionally unstable

  25. AE: Centered-Euler Neverbetween [0,1] together >>>>> Unstable!

  26. AE: DDR Unconditionally unstable!

  27. AE: phase errors Dispersion errors: 3rd order

  28. Stabilize AE centered Addartificialdiffusion Lax-Wendroff Upwind Time centered ……

  29. Stabilize AE: Addartificialdiffusion

  30. ADE: Centered-Euler

  31. ADE: Centered-Euler

  32. Stability vsRealizability

  33. Stability analysis: general Propagation

  34. Stability analysis: general (<1) (>0)

  35. General: Stability and dispersion Infrared limit: k,omega to zero

  36. ADE: Stability analysis Conditionally stable!

  37. Lax-Wendroff

  38. LW: Stability analysis Unconditionally stable!

  39. AE: Upwind Onesided: first order in BOTH space and time

  40. AE: Upwind For 0<alpha<2 botha,b are in [0,1]: Stable!

  41. Upwind: DDR

  42. Summary ========= =========== DE_centered AE_centered ADE_centered ADE_LW AE_Upwind

  43. End of lecture 6

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