Usually higher order temporal discretization is used when the spatial

Usually higher order temporal discretization is used

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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.
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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
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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.
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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
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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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