时间发展方程的交替方向显式方法及其在分数阶微分方程中的应用
An Alternating Direction Explicit Method for Time Evolution Equations with Applications to Fractional Differential Equations
摘要：
分别推导和分析了具有含时Dirichlet边界条件和零Neumann边界条件的时间发展方程的交替方向显式（ADE）方法.原始的ADE方法是一种加性算子分裂（AOS）方法，它已发展用于处理广泛的线性和非线性时间发展方程的零Dirichlet边界条件。对于线性方程，它已被证明，以实现二阶精度的时间，但对于任意时间步长是无条件稳定的。对于本文所考虑的边界条件，我们在计算区域边界附近的网格点上仔细构造了修正公式，并证明了这些公式保持了所需的精度和无条件稳定性。我们还构造了两类分数阶微分方程的基于ADE格式的数值方法。我们将给出数值例子来证明该方法的简单性和计算效率。
英文摘要：
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.
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