Discrete Artificial Boundary Conditions Research Articles

Stability and convergence analysis of artificial boundary conditions for the Schrödinger equation on a rectangular domain

Based on the discrete artificial boundary condition introduced in [23] for the two-dimensional free Schrodinger equation in a computational rectangular domain, we propose to analyze the stability and convergence rate of the resulting full scheme. We prove that the global scheme is L 2-stable and that the accuracy is second-order in time, confirming then the numerical results reported in [23].

Fast Evaluation of Artificial Boundary Conditions for Advection Diffusion Equations

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.

The finite difference scheme for nonlinear Schrödinger equations on unbounded domain by artificial boundary conditions

In this paper, we propose and analyze a finite difference method for the nonlinear Schrödinger equations on unbounded domain by using artificial boundary conditions. Two artificial boundary conditions are introduced to restrict the original Schrödinger equations on an unbounded domain into an initial–boundary value problem with a bounded domain. Then, a finite difference scheme for the reduced problem is proposed. The important feature of the proposed scheme is that an extrapolation operator is introduced to treat the nonlinear term while the scheme keeps unconditionally stable and does not introduce any oscillations at the artificial boundaries. The proposed scheme with the discrete artificial boundary conditions is rigorously analyzed to yield the unconditional stability and the scheme is also proved to be convergent. Numerical examples are given to show the performance of our scheme.

Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation

In this paper we consider two numerical scheme based on trapezoidal rule in time for the linearized KdV equation in one space dimension. The goal is to derive some suitable artificial boundary conditions for these two full discretization using Z-transformation. We give some numerical benchmark examples from the literature to illustrate our findings.

Discrete Artificial Boundary Conditions for the Lattice Boltzmann Method in 2D

To conne a spatial domain to a smaller computational domain, one needs articial bound- aries. This work considers the lattice Boltzmann method and deals with boundary conditions for these open boundaries. Ideally, such a condition does not interact with the uid at all. We present novel two-dimensional discrete articial boundary conditions to pursue that goal and we discuss four dier- ent versions. This type of condition is formulated on the discrete lattice Boltzmann level and does not require a PDE formulation of the uid. We set a special focus on the D2Q9 model. Our numeri- cal results compare the novel discrete articial boundary conditions to simulations using the existing non-reecting characteristic boundary condition and an exit boundary condition.

Concept for a one-dimensional discrete artificial boundary condition for the lattice Boltzmann method

This article deals with artificial boundaries which you encounter when a large spatial domain is confined to a smaller computational domain. Such an artificial boundary condition should not preferably interact with the fluid at all. Standard boundary conditions, e.g., a pressure or velocity condition, result in unphysical reflections. So far, existing artificial boundary conditions for the lattice Boltzmann method (LBM) are transferred from macroscopic formulations.In this work we propose novel discrete artificial boundary conditions (ABCs) which are tailored on the LBM’s mesoscopic level. They are derived directly for the chosen LBM with the aim of higher accuracy. We describe the idea of discrete ABCs in a three velocity (D1Q3) model governing the Navier–Stokes equations in one dimension. Numerical results finally demonstrate the superiority of our new boundary condition in terms of accuracy compared to previously used ABCs.

Exact artificial boundary conditions for a lattice Boltzmann method

When using a lattice Boltzmann method on an unbounded (or very large) domain one has to confine this spatial domain to a computational domain. This is realized by introducing so-called artificial boundary conditions. Until recently, characteristic boundary conditions for the Euler equations were considered and adapted to the lattice Boltzmann method.In this work we propose novel discrete artificial boundary conditions which are derived directly for the chosen lattice Boltzmann model, i.e., on the discrete level. They represent the first exact artificial boundary conditions for lattice Boltzmann methods. Doing so, we avoid any detour of considering continuous equations and obtain boundary conditions that are perfectly adapted to the chosen numerical scheme. We illustrate the idea for a one dimensional, two velocity (D1Q2) lattice Boltzmann method and show how the computational efficiency can be increased by a finite memory approach. Analytical investigations and numerical results finally demonstrate the advantages of our new boundary condition compared to previously used artificial boundary conditions.

An analysis of the finite-difference method for one-dimensional Klein–Gordon equation on unbounded domain

The numerical solution of the one-dimensional Klein–Gordon equation on an unbounded domain is analyzed in this paper. Two artificial boundary conditions are obtained to reduce the original problem to an initial boundary value problem on a bounded computational domain, which is discretized by an explicit difference scheme. The stability and convergence of the scheme are analyzed by the energy method. A fast algorithm is obtained to reduce the computational cost and a discrete artificial boundary condition (DABC) is derived by the Z-transform approach. Finally, we illustrate the efficiency of the proposed method by several numerical examples.

A FAST, STABLE AND ACCURATE NUMERICAL METHOD FOR THE BLACK–SCHOLES EQUATION OF AMERICAN OPTIONS

In this work we improve the algorithm of Han and Wu [SIAM J. Numer. Anal. 41 (2003), 2081–2095] for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank–Nicolson scheme for solving the Black–Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level. Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions. Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.

Discrete artificial boundary conditions for nonlinear Schrödinger equations

In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank–Nicholson scheme, the Dùran–Sanz-Serna scheme, the DuFort–Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.

Discrete artificial boundary conditions for transient scalar wave propagation in a 2D unbounded layered media

This paper presents an efficient numerical method for direct time-domain solution of the transient scalar wave propagation in a two-dimensional unbounded multi-layer soil. The unbounded domain is truncated by an artificial boundary which demands the corresponding boundary conditions. In the new approach, only the artificial boundary is discretized into one-dimensional finite elements, yielding a new time-dependent partial differential equation (PDE) for displacements with respect to only one spatial coordinate. Factorization of the PDE and introduction of the residual radiation functions, there results a linear first-order ordinary differential equation (ODE). Its stability is ensured. The time-dependent discrete artificial boundary conditions are determined by the solution of the ODE. In general, it is local in time, but it is non-local in space. Several numerical examples are given to verify the superiority of the proposed method.

The discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of Poisson equation by using the direct method of lines

The numerical simulation for the exterior problem of Poisson equation is considered. We introduced a polygonal artificial boundary Γ e and designed a discrete artificial boundary condition on it by using the direct method of lines. Then the original problem is reduced to a boundary value problem defined in a bounded computational domain with a polygonal boundary. The finite element approximation of this reduced boundary value problem is considered and it is proved that the finite element approximate problem is well posed. Furthermore numerical examples show that the discrete artificial boundary condition is very effective and more accurate than the Neumann boundary condition which is often used in engineering literatures.

The direct method of lines for the problem of infinite elastic foundation

We consider the numerical simulation for the problem of infinite elastic foundation. A polygonal artificial boundary is introduced and a discrete artificial boundary condition on it is presented by using the direct method of lines. Then, the original problem is reduced to a boundary value problem on a bounded computational domain, which is solved by the finite element method. In addition, we prove an optimal a priori error bound for the displacement in the bounded computational domain. Finally, numerical example shows that the discrete artificial boundary condition given in this paper is very effective and more accurate than Neumann boundary condition which is often used in engineering literatures.

Artificial Boundary Conditions for the Linearized Compressible Navier–Stokes Equations: II. The Discrete Approach

In the first part of this paper (J. Comput. Phys.137, 1, 1997), continuous artificial boundary conditions for the linearized compressible Navier–Stokes equations were proposed which were valid for small viscosities, high time frequencies, and long space wavelengths. In the present work, a new hierarchy of artificial boundary conditions is derived from the so-called “discrete” approach, which consists in working directly on the discretized equations, under the assumption of low time frequencies instead of small viscosities. The discrete artificial boundary conditions are implemented in 1D and 2D model problems and they compare quite well with the continuous artificial boundary conditions. Being self-sufficient by construction, they can be used as numerical boundary conditions and be coupled to schemes having arbitrary stencils.

Numerical simulation for the problem of infinite elastic foundation

In this paper we consider the numerical simulation for the problem of infinite elastic foundation. A half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines. Then the original problem is reduced to a boundary value problem on a bounded computational domain. Furthermore, finite element approximation of this boundary value problem defined in the bounded domain is considered. The numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the Neumann boundary condition which is often used in engineering literatures.

A discrete artificial boundary condition for steady incompressible viscous flows in a no-slip channel using a fast iterative method

We design a discrete artificial boundary condition for the steady incompressible Navier-Stokes equations in stream function vorticity formulation in an infinite channel. The new boundary condition is derived from a linearized Navier-Stokes system and a fast iterative method. Numerical experiments for the nonlinear Navier-Stokes equations are presented. The discrete artificial boundary condition is compared to Dirichlet and Neumann boundary conditions for the flow over a forward or backward facing step and for flow past a rectangular cylinder. Numerical results show that our boundary condition is very effective.