Numerical solution of the Helmholtz and the wave equations in unbounded domains.

摘要

Important areas of application in applied mathematics and engineering such as acoustic or electromagnetic scattering require a boundary condition at infinity in order to guarantee a unique and well-posed solution. To solve a boundary value problem numerically in an unbounded domain, we limit the computational domain to a finite region by introducing an artificial boundary. This requires a boundary condition on the artificial boundary such that the solution of the problem in the bounded region coincides with the solution of the original unbounded problem.; In this work, we present the numerical solution of the two dimensional time-harmonic wave (Helmholtz) equation and the time-dependent wave equation in two and three space dimensions where the domain is unbounded.; In the time-harmonic problem, we consider as an example the scattered wave problem when plane waves bombard a circular cylinder where the artificial boundary is a circle surrounding the cylinder. We examine the performance of Sommerfeld, Bayliss, and nonreflecting boundary conditions. First, on the artificial boundary, we compare the normal derivative approximated with these conditions to the exact normal derivative. Second, we calculate the numerical solution of Helmholtz problem with the second and the fourth order finite difference method. The accuracy and the rate of convergence of the numerical procedure are estimated. Another example using a point source for the Helmholtz problem, is tested and the sensitivity of the solution to the radius of the artificial boundary is discussed.; In the time-dependent problem, new exact nonreflecting boundary conditions are obtained for the two and the three dimensional wave equation and three techniques are used to derive the approximate nonreflecting conditions. The accuracy and convergence properties of these boundary conditions are tested using the normal derivative approximation and the explicit finite difference method combined with the boundary condition to calculate the numerical solution. The numerical examples show that our boundary conditions are very accurate and remain stable for a long time.