Finite Difference Schemes for ∂u∂t=(∂∂x)αδGδu That Inherit Energy Conservation or Dissipation Property
✍ Scribed by Daisuke Furihata
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 160 KB
- Volume
- 156
- Category
- Article
- ISSN
- 0021-9991
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✦ Synopsis
We propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations, such as the Korteweg-de Vries (KdV) equation and the Cahn-Hilliard equation. The most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts, which implies that the inherited properties are satisfied exactly. Since the inherited properties are kept even if the time mesh size changes in the time-evolution process, we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes. Because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to PDE solutions and sufficiently flexible to treat. The inheritance of the energy conservation and dissipation properties are verified numerically for the KdV equation and the Cahn-Hilliard equation