论文标题
时间跨度非线性次扩散方程的时间稳定数值方法的收敛分析
Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations
论文作者
论文摘要
1986年，迪克森（Dixon）和麦基（McKee）产生了一个离散的分数grönwall不平等[Z.安格。数学。 Mech。，66（1986），第535--544页，可以看作是经典离散grönwall不平等的概括。但是，这种广义的离散不平等现象并未被广泛应用于时间段进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一般的离散grönwall不平等，以证明用于时间段的非线性非线性亚次扩散方程的一类时步进数值方法的收敛性，包括流行的一阶分数分数分数式式曲柄曲柄曲柄式曲柄式式式式式式辅助式式式式式式式式曲柄 - nicolson类型方法。我们在空间离散化中获得了最佳$ l^2 $错误估计。快速时间步变的数值方法的收敛也可以简单地证明。
In 1986, Dixon and McKee developed a discrete fractional Grönwall inequality [Z. Angew. Math. Mech., 66 (1986), pp. 535--544], which can be seen as a generalization of the classical discrete Grönwall inequality. However, this generalized discrete Grönwall inequality has 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 Grönwall inequality to prove the convergence of a class of time-stepping numerical methods for time-fractional nonlinear subdiffusion equations, including the popular fractional backward difference type methods of order one and two, and the second-order fractional Crank-Nicolson type methods. We obtain the optimal $L^2$ error estimate in space discretization. The convergence of the fast time-stepping numerical methods is also proved in a simple manner.
