The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

Abstract

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of time-dependent diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. The method of lines (MOL) is used to combine spatial and temporal discretizations. The spatial scheme requires the definition of a high-order approximation of the divergence and gradient operators and the two inner products for the discrete analogs of fluxes and scalar unknowns. The discrete divergence and gradient operators are built according to a discrete duality relation. The inner product for the flux grid functions is built by explicitly imposing the conditions of consistency and stability. The family of semi-discrete mimetic methods is proved theoretically to be energy-stable as the corresponding continuous problem. Then, a full discretization is derived by combining via MOL the MFD method of order k with time marching schemes from the backward differentiation formula of order k + 2. Optimal order of accuracy is demonstrated for the scalar variable and verified numerically by solving the time-dependent diffusion problems with a variable diffusion tensor for k from 0 to 3 on three different unstructured mesh families.

Boundary treatment of linear multistep methods for hyperbolic conservation laws

“…When solving time dependent partial differential equations (PDEs), it is common to first discretize the spatial derivatives to form a system of ordinary differential equations (ODEs), and then solve the system of ODEs by time-stepping methods, such as Runge-Kutta methods and linear multistep methods (LMMs). This is the so-called method of lines (MOL) and is efficient for many time dependent PDEs [4,5,6,7,8,9,10,11,12,13,14]. When the PDEs to be solved are restricted in frequently encountered bounded domains, it is necessary to properly treat boundary conditions so that the overall accuracy and stability are maintained.…”