Monotonicity correction for second order element finite volume methods of anisotropic diffusion problems

Abstract

We apply the monotonicity correction to second-order element finite volume methods, which include quadratic element on triangular meshes and biquadratic element on quadrilateral meshes, for anisotropic diffusion problems, and obtain a second-order monotone finite volume scheme. The main ideas of monotonicity correction are as follows: When we formulate the second-order element finite volume schemes, we need to calculate line integrals on the boundary line segment of the dual element, and regard these line integrals as numerical fluxes, which are in the form of multi-point flux. Among these numerical fluxes, we can separate a two-point flux structure on both sides of the line segment. Therefore, we can apply nonlinear monotone correction to these numerical fluxes, then we obtain a corrected second order element finite volume schemes whose stiffness matrix is monotone matrix, so the first high order unconditional positivity-preserving finite volume schemes are obtained by us. Moreover, we analyze the truncation error of the corrected numerical fluxes. Furthermore, for the time dependent problem, we also apply monotone correction to time derivative term. Numerical experiments show that the corrected second order element finite volume schemes have monotonicity, and maintain the convergence order of the original schemes, which are second order in H1 norm and third order in L2 norm.