The discrete variational derivative method based on discrete differential forms
- 作者：Takaharu Yaguchi ; yaguchi@mist.i.u-tokyo.ac.jp ; Takayasu Matsuo ; Masaaki Sugihara
- 关键词：Discrete variational derivative method ; Geometric integration ; Conservation ; Dissipation ; Discrete differential form ; Discrete exterior calculus
- 刊名：Journal of Computational Physics
- 出版年：2012
- 期刊代码：136_00219991
- 类别：cp
- 出版时间：20 May, 2012
- 卷：231
- 期：10
- 页码：3963-3986
- 文件大小：2652 K
摘要
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 鈥渄iscrete variational derivative method鈥?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 H1-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 Poincar茅 inequality, which are the temporal and spacial structures that are preserved by the above methods.