SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3530-3557, January 2020.
An artificial boundary method is developed for solving the one-dimensional advection diffusion equation in the real line. In order to construct a fully discrete fast numerical algorithm with rigorous error analysis, we start with the two-step backward difference formula for time discretization of the advection diffusion equation in the whole real line. Then, we use the discrete analogue of the Laplace transform to derive a second-order time-stepping scheme in a bounded domain equipped with a discrete artificial boundary condition (ABC). The Galerkin finite element method is used for spatial discretization. To expedite the evaluation of time convolution involved in the discrete ABC, we propose a fast algorithm based on the best rational approximation of square root function in subdomains of the complex plane. An estimate for this best rational approximation enables us to prove optimal-order convergence of the fully discrete numerical scheme (integrating the fast approximation algorithm). Several numerical examples are provided to illustrate the convergence of numerical solutions and the effectiveness of the proposed fast approximation algorithm.

中文翻译：

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对流扩散方程的人工边界条件的快速评估

SIAM数值分析杂志，第58卷，第6期，第3530-3557页，2020年1月。
提出了一种人工边界方法来求解实线中的一维对流扩散方程。为了构建具有严格误差分析的全离散快速数值算法，我们从两步后向差分公式开始，对整个实线中的对流扩散方程进行时间离散。然后，我们使用Laplace变换的离散类似物在配备有离散人工边界条件（ABC）的有界域中导出二阶时间步长方案。Galerkin有限元方法用于空间离散化。为了加快对离散ABC中涉及的时间卷积的评估，我们提出了一种基于平方根函数在复平面子域中的最佳有理逼近的快速算法。对这种最佳有理逼近的估计使我们能够证明完全离散数值方案的最优阶收敛（集成了快速逼近算法）。提供了几个数值示例来说明数值解的收敛性以及所提出的快速逼近算法的有效性。