We survey some recent applications of radial basis functions (rbfs) for the BEM and related algorithms such as the method of fundamental solutions. Among these are the use of alternatives to the traditional 1+ r function in the dual reciprocity method such as thin plate splines, multquadrics and the recently discovered compactly supported positive definite rbfs, and convergence proofs of the DRM for Poisson’s equation. Newly discovered particular solutions for Helmholtz-type operators are discussed and applied to give efficient mesh free algorithms for the diffusion equation. In addition, a number of proposals are given for future applications of rbfs such as the use of surface rbfs for interpolation and the solution of boundary integral equations and the application of Kansa’s method to develop new rbf based coupled domain-boundary approximation methods.

Some recent results and proposals for the use of radial basis functions in the BEM

This paper is concerned with transparent boundary conditions 
(TBCs) for the time-dependent Schrodinger equation in one and two 
dimensions. Discrete TBCs are introduced in the numerical simulations 
of whole space problems in order to reduce the computational 
domain to a finite region. Since the discrete TBC for the Schrodinger 
equation includes a convolution w.r.t. time with a weakly decaying 
kernel, its numerical evaluation becomes very costly for large-time 
simulations. 
As a remedy we construct approximate TBCs with a kernel having 
the form of a finite sum-of-exponentials, which can be evaluated 
in a very efficient recursion. We prove stability of the resulting initialboundary 
value scheme, give error estimates for the considered approximation 
of the boundary condition, and illustrate the efficiency of 
the proposed method on several examples.

/pdf/discrete-transparent-boundary-conditions-for-the-schrodinger-2lp70pzjzr.pdf

Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability

This article reviews convolution quadrature and its uses, extends the known approximation results for the case of sectorial Laplace transforms to finite-part convolutions with non-integrable kernel, and gives new, unified proofs of the optimal error bounds for both locally integrable and non-integrable convolution kernels.

Convolution Quadrature Revisited

Time-dependent problems that are modeled by initial-boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary int egral equation method. The ideal situation is when the right-hand side in the partial differential equation and th e initial conditions vanish, the data are given only on the boundary of the domain, the equation has constant coefficien ts, and the domain does not depend on time. In this situation, the transformation of the problem to a boundary integral equation follows the same well-known lines as for the case of stationary or time-harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: Reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary. There are, however, specific difficulties due to the addition al time dimension: Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the elliptici ty of the underlying boundary value problem. In the timedependent case, instabilities have been observed in practi ce, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available. In this article, the mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented. We describe some of the main algorithms that are used in practice and have been analyzed in the mathematical literature.

/pdf/time-dependent-problems-with-the-boundary-integral-equation-2m1b5oa792.pdf

Time‐Dependent Problems with the Boundary Integral Equation Method

The paper deals with the retarded potential boundary integral equations (RPBIE) used in the numerical resolution of transient scattering problems (the so-called time domain boundary element methods). We propose here a review and update of the mathematical analysis of the involved RPBIE. Our approach, via Laplace transform, is described in some details for the classical acoustic scattering problems. The main results are: (i) existence and uniqueness theorems on a functional framework closely linked to the energy of the scattered waves; (ii) space-time variational formulations for the so-called “first kind” RPBIE, with coerciveness obtained by energy estimates. That leads us to advocate choosing these first kind RPBIE and their Galerkin approximations, instead of the second kind RPBIE and the collocation approximations. The actual space-time boundary elements are described in some detail. Examples of numerical experiments that confirm the unconditional stability of our schemes are reported, as well as references for similar results in other related work.

On Retarded Potential Boundary Integral Equations and their Discretisation

Convergence estimates in terms of the data are shown for multistep methods applied to non-homogeneous linear initial-boundary value problems. Similar error bounds are derived for a new class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems, e.g., for the single-layer potential equation of the wave equation. In both cases, the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions. These estimates, which are also proved here, depend on bounds of the Laplace transform of the (distributional) convolution kernel outside the stability region scaled by the time stepsize, and on the smoothness of the data.

On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations

The present paper develops the theory of general Runge-Kutta methods for Volterra integrodifferential equations. The local order is characterized in terms of the coefficients of the method. We investigate the global convergence of mixed and extended Runge-Kutta methods and give results on asymptotic error expansions. In a further section we construct examples of methods up to order 4.

Ch. Lubich

Papers

On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations

Runge-Kutta theory for Volterra integrodifferential equations