An Implicit Finite Difference Scheme with Preconditioning for Convection Dominated Diffusion Equation.
ABSTRACT An implicit finite difference scheme for solving time-dependent convection dominated diffusion equations in two space variables is presented. A one-sided difference approximation is used for the convection terms and a second-order central difference approximation for the diffusion term. The implicit scheme is consistent, unconditionally stable and itpsilas L2 error estimation is optimal. By preconditioning technique of incremental unknowns, the implicit scheme is an efficient one.
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ABSTRACT: The authors apply the method of incremental unknowns to multilevel finite difference approximations of linear second order elliptic boundary value problems. Roughly speaking, in a two-level setting the incremental unknowns consist of the nodal values at the coarse grid points and, for the fine grid points, of the increment to the averaged value a the neighbouring coarse grid points. Evidently, there is a close relationship to H. Yserentant’s hierarchical basis finite element method [Numer. Math. 49, 379-412 (1986; Zbl 0608.65065)] but it should be emphasized that the two methods are not the same. In case of the Dirichlet problem and spatial step size h it is shown that the condition number behaves like O((logh) 2 ) compared to O(h -2 ) for the standard nodal unknowns. The theoretical results are supported by several numerical examples.Siam Journal on Matrix Analysis and Applications - SIAM J MATRIX ANAL APPLICAT. 01/1993; 14(2).