Basic HTML version of Foils prepared 15 March 1996

Foil 9 Relation between Discretized time in Hyperbolic/Parabolic Equations and Iteration Index for Solution of Steady State Equs

From Spatial Differencing and ADI Solution of the NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. by Geoffrey C. Fox
1 We already showed in CPS615 how one could derive Jacobi, Gauss-Seidel and SOR Iterative schemes from Artificial Diffusion Equations.
  • Other important steady-state methods such as Conjugate Gradient are not elegantly looked at this fashion but basic idea still relevant.
2 One derives Iterative methods by splitting matrix K of steady state KU=b, into two
  • K = M - N , so that one finds M U = N U + b
  • or the iteration MU(n+1)= NU(n) + b
  • which M (U(n+1)-U(n)) = -KU(n) + b
3 The matrix M corresponds to Mim(ex)plicit in typical time dependent hyperbolic numerical solver
4 Note M is in simple cases (Jacobi) diagonal and is typically diagonally dominant like hyperbolic solver
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