Finite Difference Schemes for ∂u∂t=(∂∂x)αδGδu That Inherit Energy Conservation or Dissipation Property
Finite Difference Schemes for ∂u∂t=(∂∂x)αδGδu That Inherit Energy Conservation or Dissipation Property
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DOI：
10.1006/jcph.1999.6377
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发表时间：
1999-11
影响因子：
4.1
通讯作者：
Daisuke Furihata
中科院分区：
文献类型：
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作者：
Daisuke Furihata
**摘要**
我们提出了一种新的程序，用于通过死记硬背设计有限差分格式，这些格式从非线性偏微分方程（如Korteweg - de Vries（KdV）方程和Cahn - Hilliard方程）继承能量守恒或耗散特性。我们的程序最重要的特点是使用分部求和对变分导数进行严格离散化，这意味着所继承的特性能够精确满足。由于即使在时间演化过程中时间网格尺寸发生变化，所继承的特性仍然保持，我们可以使用一些适当的时间网格自适应方法通过推导出来的格式获得数值解。由于这些特性，推导出来的格式有望在数值上稳定，产生收敛于偏微分方程解的解，并且具有足够的灵活性以便处理。对于KdV方程和Cahn - Hilliard方程，能量守恒和耗散特性的继承通过数值方法得到了验证。
Abstract 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