Positive composite finite volume schemes for the diffusion equation on unstructured meshes
Résumé
We present a finite volume scheme for the anisotropic diffusion equation. This scheme is obtained as a limit of an asymptotic preserving (AP) scheme for the M1 model of radiative transfer. The latter was designed in [BDF12] and [Fra12] on polygonal meshes and [BHL21] on conical meshes. After having presented the construction of the scheme, we show that it writes as a convex combination of two consistent terms. The first one is second order consistent and may generate instabilities on unstructured meshes. The second one is first order consistent and more stable. It can be modified so as to reach a second order consistency using a reconstruction procedure. Moreover, we prove that the explicit time discretisation of our scheme preserves the positivity of the unknown under a CFL condition. Some numerical test cases are given in order to illustrate the good properties of the scheme.
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