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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. [ABSTRACT FROM AUTHOR]

The stability and error analysis of a second-order fast approximation are considered for the one-dimensional local and nonlocal diffusion equations in the unbounded spatial domain. We first use the conventional central difference scheme to discretize the local second-order spatial derivative operator and use an asymptotically compatible difference scheme to discretize the spatial nonlocal diffusion operator, and apply second-order backward differentiation formula (BDF2) to approximate the temporal derivative to achieve a fully discrete infinity system. To solve the resulting fully discrete systems, we develop a unified framework that is applicable to the discretization of both local and nonlocal problems. A key ingredient is to derive Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs). To do so, we apply the z-transform and solve an exterior problem using an iteration technique to derive a Dirichlet-to-Dirichlet (DtD)-type mapping as exact ABCs. After that, we use the Green formula to reformulate the DtD-type mapping equivalently as the DtN-type mapping. The resulting DtN-type mapping allows us to reduce the infinity discrete system into a finite discrete system in a truncated computational domain of interest, and also make it possible to present the stability and convergence analysis of the reduced problem under some open but reasonable assumptions. To efficiently implement the exact ABCs, we further develop a fast convolution algorithm based on approximation of the contour integral induced by the inverse z-transform. The stability and error analysis of the reduced finite discrete system based on the fast algorithm for exact ABCs are also established, and numerical examples are provided to demonstrate the effectiveness of our proposed approach. [ABSTRACT FROM AUTHOR]

Implicit Euler approximations of the equations governing the porous flow of two immiscible incompressible fluids are shown to be the Euler-Lagrange equations of a convex function. Tools from convex analysis are then used to develop robust fully discrete algorithms for their numerical approximation. Existence and uniqueness of solutions to control volume approximations are established. [ABSTRACT FROM AUTHOR]