| 守恒型扩散方程非线性离散格式的性质分析和快速求解 |
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| 引用本文： | 崔霞,岳晶岩.守恒型扩散方程非线性离散格式的性质分析和快速求解[J].计算数学,2015,37(3):227-246. |
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| 作者姓名： | 崔霞 岳晶岩 |
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| 作者单位： | 北京应用物理与计算数学研究所计算物理实验室, 北京 100088 |
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| 基金项目： | 国家自然科学基金(11171036, 11271054, 11301033), 中国工程物理研究院科学技术发展基金(2012B0202026, 2014A0202010), 计算物理实验室基金 |
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| 摘 要： | 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.
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| 关 键 词： | 守恒型扩散方程 非线性全隐离散格式 二阶时间精度 存在唯一性 迭代加速 |
| 收稿时间： | 2014-12-10; |
Property analysis and quick solutions for nonlinear discrete schemes for conservative diffusion equation |
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| Cui Xia,Yue Jingyan.Property analysis and quick solutions for nonlinear discrete schemes for conservative diffusion equation[J].Mathematica Numerica Sinica,2015,37(3):227-246. |
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| Authors: | Cui Xia Yue Jingyan |
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| Institution: | National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China |
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| Abstract: | 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. |
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| Keywords: | Conservative diffusion problem nonlinear fully implicit discrete scheme second-order temporal accuracy unique existence iteration acceleration |
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