Lubich convolution quadratures and their application to problems described by space-time BIEs

Abstract

In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.

Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

“…Here we wish to show a similar result for non-sectorial functions K (s, d) of the previous section. The result will only hold for large enough j and has already been stated in [14] as a conjecture based on numerical experiments. In order to simplify the presentation, we set for the rest of the paper the speed of propagation of waves to c = 1.…”

Exact non-reflecting boundary condition for 3D time-dependent multiple scattering–multiple source problems

“…Nevertheless, in our opinion there is still room for reducing these drawbacks, including the computational complexity and the working space. This includes the use of discrete convolution quadratures alternative to those of Lubich (see [31,26,32]), which should allow the construction of highly sparse K m and V m matrices, with the position of the non zero elements known a priori, but at the cost of losing the FFT benefits; the use of higher order Lubich convolution rules (see for example [21,33]) or of time integration formulas which do not require to fix a priori the final time instant T and to proceed with constant time step-size (see [34,35]). Finally, we recall that very recently data-sparse techniques, such as panel-clustering, H-matrices and highfrequency fast multipole methods, have been used to reduce the overall computational cost of a 3D space-time BIE Galerkin discretization (see [36,37]).…”

“…Note that, contrary to the 2D case, where the corresponding kernels K J (r, s) have a weak singularity at s = 0 (see Section 3.1 in [20], in particular Remark 3.1), for r > 0 the above ones are (analytic) entire functions with respect to the variable s. Furthermore, the associated convolution coefficients ω J n have a more favorable behavior (see [21]), which gives rise to more sparse matrices, whose elements actually decay to zero as the time step-size tends to zero (see Section 3.4). Also the computation of the matrix elements (16) and (17) below benefits from this behavior.…”

“…In the case of the operator V, the following explicit representation for the ω coefficients has been derived in [24] (see also [21]):…”

Time-domain boundary integral equation modeling of heat transmission problems

“…A multistage version of the method was derived only a couple of years later in [28]. CQ techniques are now widely used in the realm of TDBIE, especially for wave propagation phenomena [34,8,25], but they are also useful in the context of TDBIE for parabolic problems [29,5]. The Laplace domain analysis of CQ has a black-box nature that makes it very attractive: it deals with general families of operators as long as their Laplace transforms (transfer functions) satisfy certain properties.…”