论文标题
与分数微分方程应用的时间演变方程的交替方向显式方法
An Alternating Direction Explicit Method for Time Evolution Equations with Applications to Fractional Differential Equations
论文作者
论文摘要
我们得出并分析具有时间依赖性的dirichlet边界条件和零neumann边界条件的时间演化方程的交替方向（ADE）方法。原始ADE方法是一种加性操作员分裂（AOS）方法，该方法已开发用于处理零dirichlet边界条件的各种线性和非线性时间演变方程。对于线性方程，已证明它可以在时间上达到二阶精度，但对于任意时间步长而言是无条件稳定的。对于这项工作中考虑的边界条件，我们在计算域边界附近的网格点上仔细构建更新公式，并表明这些公式保持了所需的准确性和无条件稳定性的属性。我们还基于ADE方案构建数值方法，用于两类分数微分方程。我们将提供数值示例，以证明该方法的简单性和计算效率。
We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive operator splitting (AOS) method, which has been developed for treating a wide range of linear and nonlinear time evolution equations with the zero Dirichlet boundary condition. For linear equations, it has been shown to achieve the second order accuracy in time yet is unconditionally stable for an arbitrary time step size. For the boundary conditions considered in this work, we carefully construct the updating formula at grid points near the boundary of the computational domain and show that these formulas maintain the desired accuracy and the property of unconditional stability. We also construct numerical methods based on the ADE scheme for two classes of fractional differential equations. We will give numerical examples to demonstrate the simplicity and the computational efficiency of the method.
