A Monotone Finite Volume Scheme with Fixed Stencils for 3D Heat Conduction Equation

In this paper, a new nonlinear finite volume scheme preserving positivity for three-dimensional (3D) heat conduction equation is proposed. Being different from the traditional monotone schemes, the flux on each 3D non-planar cell-face is entirely approximated by the so-called effective directional flux firstly, then the effective directional flux is decomposed by the fixed stencils. Fixed stencil means the decomposition is just conducted on this face such that searching the convex decomposition stencil over all cell-faces is avoided. This feature makes our scheme more efficient than the traditional monotone ones based on the adaptive stencils for convex decompositions, especially in 3D. In addition, similar to other schemes based on the fixed stencils, there is also no assumption of the non-negativity of the interpolated cell-vertex unknowns. Some benchmark examples are presented to demonstrate the second-order accuracy. Two anisotropic diffusion problems show that not only can our schemes maintain the positivity-preserving property, but also they are more efficient than the scheme based on the adaptive stencil.