The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems
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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.
KeywordsMimetic finite difference method Polygonal mesh High-order discretization Time-dependent diffusion problem Mixed formulation
Mathematics Subject Classification65M06 65M12 65M20 76R99
This work was partially supported by the Laboratory Directed Research and Development program (LDRD), under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy (DOE) at Los Alamos National Laboratory operated by Los Alamos National Security LLC under Contract No. DE-AC52-06NA25396, and the Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics of the DOE Office of Science. The University of Padova “Project SID-2016- Approximation and discretization of PDEs on Manifolds for Environmental Modeling” is also acknowledged.
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