Potential evaluation in space–time BIE formulations of 2D wave equation problems

Abstract

In this paper we examine the computation of the potential generated by space–time BIE representations associated with Dirichlet and Neumann problems for the 2D wave equation. In particular, we consider the efficient evaluation of the (convolution) time integral that appears in the potential representation. For this, we propose two simple quadrature rules which appear more efficient than the currently used ones. Both are of Gaussian type: the classical Gauss–Jacobi quadrature rule in the case of a Dirichlet problem, and the classical Gauss–Radau quadrature rule in the case of a Neumann problem. Both of them give very accurate results by using a few quadrature nodes, as long as the potential is evaluated at a point not very close to the boundary of the PDE problem domain. To deal with this latter case, we propose an alternative rule, which is defined by a proper combination of a Gauss–Legendre formula with one of product type, the latter having only 5 nodes.The proposed quadrature formulas are compared with the second order BDF Lubich convolution quadrature formula and with two higher order Lubich formulas of Runge–Kutta type. An extensive numerical testing is presented. This shows that the new proposed approach is very competitive, from both the accuracy and efficiency points of view.