• Usually, higher-order temporal discretization is used when the spatial discretization is also of higher order. • Stability : A discretization method is said to be stable if it does not magnify the errors that appear in the course of numerical solution process. • Pre-conditioning method is used when the matrix of the linear algebraic system is ill-posed, such as multi- phase flows, flows with a broad range of Mach numbers, etc. • Selection of discretization methods should consider efficiency, accuracy and special requirements, such as shock wave tracking.
21 Discretization methods (example) 0 y v x u 2 2 y u e p x y u v x u u • 2D incompressible laminar flow boundary layer m=0 m=1 L-1 L y x m=MM m=MM+1 (L,m-1) (L,m) (L,m+1) (L-1,m) 1 l l l m m m u u u u u x x 1 l l l m m m v u v u u y y 1 l l l m m m v u u y FD Sign( )<0 l m v l m v BD Sign( )>0 2 1 1 2 2 2 l l l m m m u u u u y y 2 nd order central difference i.e., theoretical order of accuracy P kest = 2. 1 st order upwind scheme, i.e., theoretical order of accuracy P kest = 1
22 Discretization methods (example) 1 1 2 2 2 1 2 1 l l l l l l l m m m m m m m FD u v v y v u FD u BD u x y y y y y BD y 1 ( / ) l l l m m m u u p e x x B 2 B 3 B 1 B 4 1 1 1 2 3 1 4 / l l l l l m m m m m B u B u B u B u p e x 1 4 1 1 2 3 1 1 2 3 1 2 3 1 2 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 l l l l l mm l mm mm p B u B B x e u B B B B B B B B u p B u x e Solve it using Thomas algorithm To be stable, Matrix has to be Diagonally dominant.
23 Solvers and numerical parameters • Solvers include: tridiagonal, pentadiagonal solvers, PETSC solver, solution-adaptive solver, multi-grid solvers, etc. • Solvers can be either direct (Cramer’s rule, Gauss elimination, LU decomposition) or iterative (Jacobi method, Gauss-Seidel method, SOR method) • Numerical parameters need to be specified to control the calculation. • Under relaxation factor, convergence limit, etc. • Different numerical schemes • Monitor residuals (change of results between iterations) • Number of iterations for steady flow or number of time steps for unsteady flow • Single/double precisions
24 Numerical methods (grid generation) • Grids can either be structured (hexahedral) or unstructured (tetrahedral). Depends upon type of discretization scheme and application • Scheme
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- Summer '18