In this lecture the performance of various solvers is analyzed when applied to 4 test cases representative of equations encountered in CFD.
The first test case deals with solution of Laplace equation in two dimensions. For this test case an analytical solution exists, so that iteration errors can be determined at each iteration. The same problem is then studied in three dimensions in the second test case.
The third test case is devoted to solving Poisson equation in two dimensions. Pressure or pressure-correction equations, which play an important role in algorithms for solving the Navier-Stokes equations for incompressible fluids, are of Poisson-type.
Finally, test case 4 deals with equation systems resulting from discretization of a transport equation for a generic scalar variable. The test case studied in lectures 6 and 9 is used, because the velocity field is available in analytic form, u = x and v = -y. The flux-making velocity is thus constant over each control volume face.
Eight solvers are tested - not all in each test case:
- Gauß-Seidel solver,
- Alternating Line Gauß-Seidel solver (i.e., line-by-line application of the direct solver for tridiagonal matrixes, TDMA, along one grid line),
- ILU solver by Stone ("strongly implicit procedure", SIP),
- Incomplete Cholesky preconditioned conjugate gradient solver (ICCG),
- CGSTAB solver,
- Multigrid acceleration with Gauß-Seidel solver as “smoother”,
- Multigrid acceleration applied to ILU solver by Stone,
- Multigrid acceleration applied to ICCG.
Computations are performed with computer codes written for this purpose; they will be available in source form in the supplementary material for this lecture.
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