守恒型扩散方程非线性离散格式的性质分析和快速求解

PROPERTY ANALYSIS AND QUICK SOLUTIONS FOR NONLINEAR DISCRETE SCHEMES FOR CONSERVATIVE DIFFUSION EQUATION

对于守恒型扩散方程,研究其二阶时间精度非线性全隐有限差分离散格式的性质,证明了其解的存在唯一性.研究了二阶时间精度的Picard-Newton迭代格式,证明了迭代解对原问题真解的二阶时间和空间收敛性,以及对非线性离散解的二次收敛速度,实现了非线性问题的快速求解.本文中方法也适用于一阶时间精度格式的分析,并可推广至对流扩散问题.数值实验验证了二阶时间精度Picard-Newton迭代格式的高精度和高效率.

Property analysis is given for nonlinear fully implicit （FI） finite difference discrete scheme with second-order time evolution for conservative diffusion equation. It is proved there exists a unique solution for the nonlinear FI scheme. A Picard-Newton iteration scheme with second-order time accuracy is studied. It is proved the solution of the iteration has second-order convergence both in spatial and temporal variants to the solution of the original problem, and it converges to the solution of the nonlinear discrete scheme with a quadratic speed. The quick solution of the nonlinear problem is realized. The methods here also adapt to analyze first-order time accurate scheme, and can be extended to convection-diffusion problem. Numerical tests verify the high accuracy and efficiency of the second-order temporal evolution Picard-Newton iteration.