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