A stable and conservative finite difference scheme for the Cahn-Hilliard equation
A stable and conservative finite difference scheme for the Cahn-Hilliard equation
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DOI：
10.1007/pl00005429
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发表时间：
2001-02
影响因子：
2.1
通讯作者：
Daisuke Furihata
中科院分区：
文献类型：
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作者：
Daisuke Furihata
我们提出了一个稳定的和保守的有限差分格式数值求解Cahn-Hilliard方程描述的相分离现象。该方程是一个非线性的近似不适定问题，很难得到数值解。我们设计了一个新的差分格式的基础上提出的一般策略Furihata和Mori。新方案继承了方程的特征性质，即质量守恒和总能量的减少。总能量的减小意味着解的离散Sobolev范数有界。这反过来又意味着，通过离散化的Sobolev引理，解的最大范数的有界性，因此解的稳定性。得到了解的误差估计，阶为。数值算例验证了该方法的有效性。
We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is. Numerical examples demonstrate the effectiveness of the proposed scheme.