Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

Zhang Hui; Zeng Fanhai; Jiang XiaoyunKarniadakis George Em

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Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

机译：时间分数阶非线性子扩散方程的时间步长数值方法的收敛性分析

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In 1986, Dixon and McKee (Z Angew Math Mech 66:535-544, 1986) developed a discrete fractional Gronwall inequality, which can be seen as a generalization of the classical discrete Gronwall inequality. However, this generalized discrete Gronwall inequality and its variant (Al-Maskari and Karaa in SIAM J Numer Anal 57:1524-1544, 2019) have not been widely applied in the numerical analysis of the time-stepping methods for the time-fractional evolution equations. The main purpose of this paper is to show how to apply the generalized discrete Gronwall inequality to prove the convergence of a class of time-stepping numerical methods for timefractional nonlinear subdiffusion equations, including the popular fractional backward difference type methods of order one and two, and the fractional Crank-Nicolson type methods. We obtain the optimal L 2 error estimate in space discretization for multidimensional problems. The convergence of the fast time-stepping numerical methods is also proved in a simple manner. The present work unifies the convergence analysis of several existing time-stepping schemes. Numerical examples are provided to verify the effectiveness of the present method.

机译：1986 年，Dixon 和 McKee （Z Angew Math Mech 66：535-544， 1986）提出了一个离散分数格朗沃尔不等式，可以看作是经典离散格朗沃尔不等式的推广。然而，这种广义离散Gronwall不等式及其变体（Al-Maskari和Karaa in SIAM J Numer Anal 57：1524-1544,2019）尚未广泛应用于时间分数阶演化方程的时间步长方法的数值分析。本文的主要目的是展示如何应用广义离散 Gronwall 不等式来证明一类时间阶乘非线性子扩散方程的时间步进数值方法的收敛性，包括流行的一阶和二阶分数阶反向差分类型方法，以及分数阶 Crank-Nicolson 类型方法。我们得到了多维问题空间离散化的最优L 2误差估计。快速时间步进数值方法的收敛性也得到了简单的证明。本文统一了几种现有时间步长方案的收敛分析。通过数值算例验证了该方法的有效性。

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《Fractional Calculus and Applied Analysis》 |2022年第2期|453-487|共35页
作者
Zhang Hui; Zeng Fanhai; Jiang XiaoyunKarniadakis George Em;
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作者单位

Shandong Univ;

Brown Univ;

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原文格式 PDF
正文语种英语
中图分类微积分;
关键词
time-fractional nonlinear subdiffusion equations; discrete fractional Gronwall inequality; fast time-stepping methods; convergence analysis; FINITE-DIFFERENCE METHODS; CONVOLUTION QUADRATURE; DIFFUSION EQUATION; ERROR ANALYSIS; SCHEME; ALGORITHMS; DISCRETIZATION; APPROXIMATIONS; MESHES;

机译：时间分数阶非线性子扩散方程;离散分数阶 Gronwall 不等式;快速时间步进方法;收敛性分析;有限差分法;卷积正交;扩散方程;错误分析;方案;算法;离散化;近似值;网格;

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Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

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