Related papers: A composite spectral scheme for variable coefficie…
In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with…
We develop a new dispersion minimizing compact finite difference scheme for the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly developed ray theory for difference equations. A discrete Helmholtz operator and a…
We develop and analyze a new approach for simultaneously computing multiple solutions to the Helmholtz equation for different frequencies and different forcing functions. The new Multi-Frequency WaveHoltz (MFWH) algorithm is an extension of…
In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth…
Time-harmonic solutions to the wave equation can be computed in the frequency or in the time domain. In the frequency domain, one solves a discretized Helmholtz equation, while in the time domain, the periodic solutions to a discretized…
We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave…
High-precision numerical scheme for nonlinear hyperbolic evolution equations is proposed based on the spectral method. The detail discretization processes are discussed in case of one-dimensional Klein-Gordon equations. In conclusion, a…
We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized…
We present and analyze fully discrete Nystr\"om methods for the solution of three classes of well conditioned boundary integral equations for the solution of two dimensional scattering problems by homogeneous dielectric scatterers.…
This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single…
In this paper we develop a plane wave type method for discretization of homogeneous Helmholtz equations with variable wave numbers. In the proposed method, local basis functions (on each element) are constructed by the geometric optics…
We describe an accelerated direct solver for the integral equations which model acoustic scattering from curved surfaces. Surfaces are specified via a collection of smooth parameterizations given on triangles, a setting which generalizes…
A fast direct inversion scheme for the large sparse systems of linear equations resulting from the discretization of elliptic partial differential equations in two dimensions is given. The scheme is described for the particular case of a…
An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal…
A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential…
We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, $u(\x) - \alpha^2 \Delta u(\x) = 0$, in bounded or unbounded multiply-connected domains. We consider both Dirichlet…
We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nystr\"om discretizations for the solution of the ensemble of associated Laplace domain…
A fast method is proposed for solving the high frequency Helmholtz equation. The building block of the new fast method is an overlapping source transfer domain decomposition method for layered medium, which is an extension of the source…
Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the…
Discretizing Helmholtz problems via finite elements yields linear systems whose efficient solution remains a major challenge for classical computation. In this paper, we investigate how variational quantum algorithms could address this…