A Monotone Finite Volume Scheme with Fixed Stencils for 3D Heat Conduction Equation (Communications in Computational Physics, 2019)
In this paper, a new nonlinear finite volume scheme preserving positivityfor three-dimensional (3D) heat conduction equation is proposed. Being different fromthe traditional monotone schemes, the flux on each 3D non-planar cell-face is entirelyapproximated by the so-called effective directional flux firstly, then the effective directional flux is decomposed by the fixed stencils. Fixed stencil means the decompositionis just conducted on this face such that searching the convex decomposition stencilover all cell-faces is avoided. This feature makes our scheme more efficient than thetraditional monotone ones based on the adaptive stencils for convex decompositions,especially in 3D. In addition, similar to other schemes based on the fixed stencils, thereis 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 thepositivity-preserving property, but also they are more efficient than the scheme basedon the adaptive stencil.
Keywords: Anisotropic diffusion tensor; distorted mesh; positivity-preserving; fixed stencils.