Vertex-Centered Linearity-Preserving Schemes for Nonlinear Parabolic Problems on Polygonal Grids

Abstract

On arbitrary polygonal grids, a family of vertex-centered finite volume schemes are suggested for the numerical solution of the strongly nonlinear parabolic equations arising in radiation hydrodynamics and magnetohydrodynamics. We define the primary unknowns at the cell vertices and derive the schemes along the linearity-preserving approach. Since we adopt the same cell-centered diffusion coefficients as those in most existing finite volume schemes, it is required to introduce some auxiliary unknowns at the cell centers in the case of nonlinear diffusion coefficients. A second-order positivity-preserving algorithm is then suggested to interpolate these auxiliary unknowns via the primary ones. All the schemes lead to symmetric and positive definite linear systems and their stability can be rigorously analyzed under some standard and weak geometry assumptions. More interesting is that these vertex-centered schemes do not have the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes (Lipnikov et al. in J Comput Phys 305:111---126, 2016). Numerical experiments are also presented to show the efficiency and robustness of the schemes in simulating nonlinear parabolic problems.