A Positivity-Preserving Finite Volume Scheme with Least Square Interpolation for 3D Anisotropic Diffusion Equation

In this paper, a new nonlinear finite volume scheme preserving positivity for 3D anisotropic diffusion equation is proposed. A distinct feature of our new scheme is that the auxiliary vertex unknowns appearing in the nonlinear two-point flux approximation are interpolated by the least-square method combined with the so-called correction function. The correction function is local, and only solved when the coefficient discontinuity occurs. Compared with other existing 3D interpolation formulas, our interpolation formula is designed for the general polyhedron mesh and coefficient discontinuity with lower and adaptive computational complexity. The scheme is proven to be positivity-preserving. Numerical examples are presented to demonstrate the second-order accuracy and positivity-preserving property for various anisotropic diffusion problems on the distorted meshes.