Class outline / SOLVING NONLIN COUPLED PDES (MTH_654_X010_F2024) CLASS SCHEDULE Fall 2024 MTH 654/9 == week 0 1. 9/25 Introduction. See F24_MTH654_Syllabus.pdf and RESOURCES 2. 9/27 Stability for schemes for linear ODE/PDE is not the same as nonlinear ODE/PDE. Example of u'+f(u)=0 when f(u) is a gradient of energy. Even if we do not have linear stability, we can have energy/gradient stability. What we want: accuracy (but this needs regularity of solutions), stability (in some sense), and solvability/robustness of solver. == week 1 3. 9/30 In general, we want schemes to produce solutions with qualitative properties similar to those of the PDE. When f(u)=u^3-u, we can show energy stability of ODE. BE, FE are not energy stable, but convexisty splitting is. For linear heat equation, we have stability. We also can show linear stability of BE but only conditional stability of FE. For nonlinear problems, we do not have stability in general. For Cahn-Allen pbm, one can only show energy stability of convexity split schemes. Solvability and solver robustness: we will discuss solving nonlinear equaitons F(x)=0 and x=G(x). 4. 10/2 Nonlinear solvers; solvability, theory, and how to use class notes (link in RESOURCES) 5. 10/4 Examples on how to use theory for Newton's method. == week 2 6. 10/7 How to test convergence order of an iterative method for solving nonlinear equations. Variants of Newton's method: secant, Shamanskii, quasi-Newton, inexact Newton. 7. 10/9 How to write code with nonuniform time stepping. How to adapt time step for reasons of accuracy (Richardson extrapolation). Why FV (and not FD, FE,...). Introduction to spatial discretization with FV on quadrangles of -(ku_x)_x=f. 8. 10/11 Details of FV approximation to -(ku_x)_x=f == week 3 9. 10/14 How to choose nonuniform grid... i.e. how to do grid adaptivity guided by a-posterior error estimates. (Not rigorously in this class). Why harmonic averages and what are the underlying approximating polynomials: motivate the choice (in 1d) of piecewise constants for scalar unknowns and piecewise linears for fluxes, and the choice of weighted harmonic averages for transmissibilities. 10. 10/16 Structural stability can mean a lot of different properties of a scheme (that hold for the true solution): maximum principle, (locally or globally) conservative fluxes, energy stability, positivity. ... Example why BE, FE, CS might not be stable in the 'usual' sense for the 0-dimensional Cahn-Allen eqn. Outline why CS (convexity splitting) is energy stable for that problem. See EYRE.pdf for other CS splitting schemes and a more general proof. 11. 10/18 Moving on to more complex PDEs: stationary and non-stationary diffusion-reaction equation. FV discretization and solver for the stationary case. Spectral properties of the "discrete diffusion" operator using Gershgorin circle theorem and analogy to the spectrum of "-d^2/dx^2".. Similarity to the properties of the discrete diffusion matrix in FV case (except for the scaling). How does coefficient k change the eigenvalues. Conditioning of the linear system. == week 4 12. 10/21 FV discretization for non-stationary reaction-diffusion. Variants of time stepping. How to set up nonlinear solver. 13. 10/23 Operator splitting (fractional time-stepping). 14. 10/25 Handout: examples how to set up and analyze an iteration for a coupled system. == week 5 15. 10/28: How to set-up/analyze simultaneous or sequential iteration for a system (handout). Solving scalar conservation law in 1d: preliminaries. 16. 10/30: Notes on schemes for conservation laws. 17. 11/1: Continue first order conservation laws: Riemann problem. Recall numerical schemes based on FD for advection equation: form, stability. == week 6 18. 11/4: Schemes for conservation laws: conservation form of the PDE motivates conservative schemes. Steps of FV technique: construct cell averages, evolve them (advect) with appropriate numerical flux to find the new cell average; reconstruct cell solution from the cell average. Construction of numerical flux from Riemann solvers (exact and approximate). Choice of numerical flux: e.g., Godunov scheme. CFl condition prevents interaction of waves. 19. 11/6: Convergence of Fv schemes; why L^1 is forgiving, why L^inf is the least forgiving. Move to flow pbms in 2D. Different viscous flow models. 20. 11/8: A2 assignment review. See worksheets W1-1.pdf and W2.pdf == week 7 21. 11/13: Lecture 21 on FLOW. 22. 11/15: Lecture 22 on TRANSPORT. == week 8 23. 11/18: recap FLOW (Darcy) and TRANSPORT (advection & diffusion/dispersion). How to handle coupled FLOW & TRANSPORT. 24. 11/20: revisitng the "coupled FLOW and TRANSPORT" issues. Demo how my code TRANSPORT2d works. 25. 11/22: discussion of the structure of TRANSPORT1d and how it can be changed to TRANSPORT2d. First discussion of Multiphysics within the Domain Decomposition framework. == week 9 26. 11/25: revisit the discussion of modeling error for transport equation (role of non-divergence free flux field). Domain decomposition : PDE setting and discrete problem for Poisson's pbm. 27. 11/27 == week 10 28. 12/2: Domain decomposition: view of solving the PDE via DD using Green's operators and harmonic extensions. Overlapping DD: Schwarz alternating procedure (additive and multiplicative). Mortar methods for multi-numerics, and multiphysics. HW B due. 29. 12/4: draft HW C2 due. 30. 12/6: REVIEW/RECAP == FINALS week 12/11: final draft/corrections of C2 due.