A Positivity-Preserving Finite Volume Scheme for Heat Conduction Equation on Generalized Polyhedral Meshes (Communications in Computational Physics, 2018)
In this paper we present a nonlinear finite volume scheme preservingpositivity for heat conduction equations. The scheme uses both cell-centered and cell-vertexunknowns. The cell-vertex unknowns are treated as auxiliary ones and areeliminated by our newly developed second-order explicit interpolation formula ongeneralized polyhedral meshes. With the help of the additional parameters, it is notnecessary to choose the stencil adaptively to obtain the convex decomposition of theco-normal vector and also is not required to replace the interpolation formula withpositivity-preserving but usually low-order accurate ones whenever negative interpolatedauxiliary unknowns appear. Moreover, the new flux approximation has a fixedstencil. These features make our scheme more efficient compared with other existingmethods based on Le Potier's nonlinear two-point approximation, especially in 3D.Numerical experiments show that the scheme maintains the positivity of the continuoussolution and has nearly second-order accuracy for the solution on the distortedmeshes where the diffusion tensor may be anisotropic and discontinuous.
Keywords: Heat conduction; anisotropic diffusion tensor; distorted mesh; positivity-preserving; cell-centered scheme; vertex elimination.