Convergence of a monotone nonlinear DDFV scheme for degenerate parabolic equations
Résumé
In this work, we carry out the convergence analysis of an efficient DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximate flux is split into two terms corresponding to the primal normal and the dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way with the help of an upwind scheme. Therefore the discrete maximum principle is respected and a uniform bound on the discrete velocity is established. As a consequence, the scheme has an approximate solution. Thanks to some compactness arguments, this discrete solution converges to that of the continuous problem provided the size of the mesh tends to zero. Some numerical tests are then presented and they show that the scheme in question turns out to be robust and efficient with an accuracy of second order.
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