Journal of Computational Physics
Journal of Computational Physics
Finite-difference schemes for nonlinear wave equation that inherit energy conservation property
Journal of Computational and Applied Mathematics
Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals
SIAM Journal on Numerical Analysis
Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes
SIAM Journal on Numerical Analysis
New conservative schemes with discrete variational derivatives for nonlinear wave equations
Journal of Computational and Applied Mathematics
A Higher-Order Formulation of the Mimetic Finite Difference Method
SIAM Journal on Scientific Computing
An extension of the discrete variational method to nonuniform grids
Journal of Computational Physics
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As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice. Lately, Furihata and Matsuo have developed the so-called ''discrete variational derivative method'' that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems. On the other hand, the theories of discrete differential forms have received much attention recently. These theories provide a discrete analogue of the vector calculus on general meshes. In this paper, we show that the discrete variational derivative method and the discrete differential forms by Bochev and Hyman can be combined. Applications to the Cahn-Hilliard equation and the Klein-Gordon equation on triangular meshes are provided as demonstrations. We also show that the schemes for these equations are H^1-stable under some assumptions. In particular, one for the nonlinear Klein-Gordon equation is obtained by combination of the energy conservation property and the discrete Poincare inequality, which are the temporal and spacial structures that are preserved by the above methods.