Generalized plane waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz pro...
Abstract We extend the nonconforming Trefftz virtual element method introduced in [30] to the case of the fluid-fluid interface problem, that is, a Helmholtz problem with piecewise constant wave number. With respect to the original approach, we address two additional issues: firstly, we define the coupling of local approximation spaces with piecewise constant wave numbers; secondly, we enrich such local spaces with special functions capturing the physical behavior of the solution to the target p...
#1Long Yuan(SDUST: Shandong University of Science and Technology)H-Index: 1
Abstract In this paper we first consider a special class of nonhomogeneous and anisotropic Helmholtz equation with variable coefficients and the source term. Combined with local spectral element method, a generalized plane wave discontinuous Galerkin method for the discretization of such Helmholtz equation is designed. Then we define new generalized plane wave basis functions for two-dimensional anisotropic Helmholtz equation with the variable coefficient. Besides, the error estimates of the app...
We introduce a novel virtual element method (VEM) for the two-dimensional Helmholtz problem endowed with impedance boundary conditions. Local approximation spaces consist of Trefftz functions, i.e....
We present asymptotic methods for solving high frequency Helmholtz equations in anisotropic media. The methods are motivated by Babich’s expansion that uses Hankel functions of the first kind to approximate the solution of high frequency Helmholtz equation in isotropic media. Within Babich’s expansion, we can derive the anisotropic eikonal equation and a recurrent system of transport equations to determine the phase and amplitude terms of the wave, respectively. In order to reconstruct the wave ...
Abstract Numerical solutions to boundary value problems governed by two-dimensional Helmholtz equation for anisotropic media is obtained. The standard BEM has been employed to obtain the solutions. The results show that the anisotropy of the medium under consideration causes effects on the solution. The anisotropy of the medium should be taken into account for the implementation of the modeling and computation.
#1Zedong Wu(KAUST: King Abdullah University of Science and Technology)H-Index: 12
#2Tariq Alkhalifah(KAUST: King Abdullah University of Science and Technology)H-Index: 33
Abstract Numerical simulation of the acoustic wave equation in either isotropic or anisotropic media is crucial to seismic modeling, imaging and inversion. Actually, it represents the core computation cost of these highly advanced seismic processing methods. However, the conventional finite-difference method suffers from severe numerical dispersion errors and S-wave artifacts when solving the acoustic wave equation for anisotropic media. We propose a method to obtain the finite-difference coeffi...
This paper presents a fast iterative solver for the Lippmann--Schwinger equation for high-frequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solver is based on the sparsifying preconditioner [L. Ying, Multiscale Model. Simul., 13 (2015), pp. 644--660] and a domain decomposition approach similar to the method of polarized traces [L. Zepeda-Nunez and L. Demanet, J. Comput. Phys., 308 (2016), pp. 347--388.] The iterative solver has two levels, the outer level...
We present a fast direct solver for two-dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of Lippmann--Schwinger type. The principal contribution here is the development of an automatically adaptive, ...
Certain types of electro-magnetic waves propagating in a plasma can undergo a mode conversion process. In magnetic confinement fusion, this phenomenon is very useful to heat the plasma, since it permits to transfer the heat at or near the plasma center. This work focuses on a mathematical model of wave propagation around the mode conversion region, from both theoretical and numerical points of view. It aims at developing, for a well-posed equation, specific basis functions to study a wave mode c...
This paper is aimed at developing new shape functions adapted to the scalar wave equation with smooth (possibly vanishing) coefficients and investigates the numerical analysis of their interpolation properties. The interpolation is local, but high order convergence is shown with respect to the size of the domain considered. The new basis functions are then implemented in a numerical method to solve a scalar wave equation problem with a mixed boundary condition. The main theoretical result states...