Journal ArticleDOI
On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations
TLDR
In this paper, convergence and stability bounds for a class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems were derived for the single-layer potential equation of the wave equation.Abstract:
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.read more
Citations
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Journal ArticleDOI
Some recent results and proposals for the use of radial basis functions in the BEM
TL;DR: In this article, a survey of radial basis functions (rbfs) for the BEM and related algorithms such as the method of fundamental solutions is presented. And a number of proposals are given for future applications of 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.
Journal ArticleDOI
Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability
TL;DR: Stability of the resulting initialboundary value scheme is proved, error estimates for the considered approximation of the boundary condition are given, and the efficiency of the proposed method is illustrated on several examples.
Journal ArticleDOI
Convolution Quadrature Revisited
TL;DR: In this paper, the authors extended the known approximation results for the case of sectorial Laplace transforms to finite-part convolutions with non-integrable kernel, and gave new, unified proofs of the optimal error bounds for both locally integrable and nonintegrably convolution kernels.
Reference EntryDOI
Time‐Dependent Problems with the Boundary Integral Equation Method
TL;DR: The mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented in this paper, where the 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.
Book ChapterDOI
On Retarded Potential Boundary Integral Equations and their Discretisation
TL;DR: In this paper, a review and update of the mathematical analysis of the involved potential boundary integral equations (RPBIE) is presented, and 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.
References
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Book
Non-homogeneous boundary value problems and applications
TL;DR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.
Journal ArticleDOI
A special stability problem for linear multistep methods
TL;DR: The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property as discussed by the authors, and error bounds are derived which are valid under rather general conditions.
Book
Initial-boundary value problems and the Navier-Stokes equations
Heinz-Otto Kreiss,Jens Lorenz +1 more
TL;DR: The Navier-Stokes equations under initial and boundary conditions were studied in this paper, where they were shown to be incompressible in the spatially periodic case and in the constant-coefficient case.
Journal ArticleDOI
Convolution quadrature and discretized operational calculus. II
TL;DR: In this article, operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular integrals (power and logarithmic singularities, finite part integrals), multiple timescale convolution, Volterra integral equations, Wiener-Hopf integral equations.
Journal ArticleDOI
Formulation variationnelle espace‐temps pour le calcul par potentiel retardé de la diffraction d'une onde acoustique (I)
TL;DR: In this paper, a space-time variational formula for the problem of transient acoustic scattering by a free (pressure release) surface, using the retarded potential technique, was given.