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Upwind strategies for local RBF scheme to solve convection dominated problems. (English) Zbl 1403.65175
Summary: The most common strategy existing in the literature for solving convection dominated Convection-Diffusion Equations (CDE) is using central approximation to the diffusive terms and upwind approximation to the convective terms. In the present work, we propose a multiquadric local RBF based grid-free upwind $$(\mathrm{LRBF}_{-}\mathrm U)$$ scheme for solving convection dominated CDE. In this method, the entire CDE operator is discretized over the nodes in the upwind local support domain for strongly convection dominant problems. The variable (optimal) shape parameter for $$\mathrm{LRBF}_{-}\mathrm U$$ scheme has been obtained by using a local optimization algorithm developed by the authors. It has been observed that for highly convection dominated problems, the $$\mathrm{LRBF}_{-}\mathrm U$$ scheme produces stable and accurate results. The proposed scheme is also been compared with the conventional Central-Upwind combined scheme, to demonstrate its superiority in generating high accurate solutions than the latter.

##### MSC:
 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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