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Comprehensive Analysis of Evolutionary PDEs: Methods and Principles in Finite-Difference Schemes

This work delves into the discretization of evolutionary partial differential equations (PDEs), focusing on finite-difference schemes for advection, diffusion, and reaction processes in fluid dynamics. Key principles such as consistency, stability, and efficiency guide the formulation and analysis of these numerical methods. The document emphasizes the importance of recovering the continuum limit, achieving stability across varying resolutions, and ensuring causality and reversibility in simulations. It aims to provide insights into the computational density, error decay, and operational cost associated with solving complex PDEs.

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Comprehensive Analysis of Evolutionary PDEs: Methods and Principles in Finite-Difference Schemes

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  1. PDE’sDiscretization Sauro Succi

  2. Evolutionary PDE Formal: big time Formal: small time

  3. Evolutionary PDE’s Advection Diffusion Reaction Self-advection (fluids)

  4. Finite-Difference Schemes

  5. Finite-Difference Schemes

  6. The guiding principles of ComPhys Consistency Recover the continuum limit at infinite resolution (no anomaly) Accuracy Fast error decay with increasing resolution Stability/Conservativeness 1st and 2nd principle, error decay Efficiency Cost per unit update

  7. The guiding principles of ComPhys Locality Computational density independent of system size (Feynman) Causality No simultaneous interactions (Present-->Future) Reversibility No burnt-bridges doors, exact roll-back, very long time integration

  8. Jump to actualPFDE’s (with apologies to the theory-inclined)

  9. Consistency

  10. Finite-Difference Schemes

  11. Consistency Consistent ForwardEuler Centered

  12. Accuracy Reproduce poly(p) at x=xj

  13. Stability

  14. Courant Numbers Faster than light?

  15. Short/Long term instability Linear instability (early) Non-linear instability (long-term)

  16. Stability: spectral analysis SpectralDeformations:

  17. Lax equivalence Theorem Consistent schemes for well-posed Linear PDE’s are convergent ifthey are stable Stability is easier to prove than convergence!

  18. Scaling limits

  19. Transfer Operator First order in time:

  20. Efficiency Acceleration Advection Diffusion Slow diffusion

  21. Computer metrics 1 Petaflop

  22. Computer metrics

  23. Locality Differential Operators Sparse matrices

  24. Causality Present depends on past NO simultaneous dependence No inverse time depenedence

  25. Reversibility(Hamiltonian)

  26. Reversibility: Euler

  27. Reversibility: Crank-Nicolson Pade’:

  28. Now to actualPDE’s

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