Asymptotic preserving schemes on conical unstructured 2D meshes

In this article, we consider the hyperbolic heat equation. This system is linear hyperbolic with stiff source terms and satisfies a diffusion limit. Some finite volume numerical schemes have been proposed which reproduce this diffusion limit. Here, we extend such schemes, originally defined on polygonal meshes, to conical meshes (using rational quadratic Bezier curves). We obtain really new schemes that do not reduce to the polygonal version when the conical edges tend to straight lines. Moreover, these schemes can handle curved unstructured meshes so that geometric error on initial data representation is reduced and geometry of the domain is improved. Extra flux coming from conical edge (through his midedge point) has a deep impact on the stabilization when compared with the original polygonal scheme. Cross-stencil phenomenon of polygonal scheme has disappeared, and issue of positivity for the diffusion problem (although unresolved on distorted mesh and/or with varying cross-section) has been in some sense improved.

Automated summary

In this article, the hyperbolic heat equation with rational quadratic Bezier curves on conical meshes is considered. New numerical schemes that handle curved unstructured meshes are developed, improving domain geometry. The impact of extra flux from conical edges on stabilization is significant compared to the original polygonal scheme. Additionally, the issues of cross-stencil phenomenon and positivity for the diffusion problem on distorted meshes have been somewhat resolved.