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 pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a sixth order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes in the literature, particularly in the critical pre-asymptotic region where $\textsf{k} h$ is near $1$ with $\textsf{k}$ being the wavenumber and $h$ the mesh size.
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This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schr\"odinger equation in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using previous mathematical results on the Schr\"odinger equation, we prove the existence of a unique solution to the Helmholtz source problem with outgoing conditions in such waveguides. We provide an explicit modal approximation of this solution, as well as a control of the approximation error in H1loc. The main theorem is proved in the case of a waveguide with a monotonously varying profile and then generalized using a matching strategy. We finally validate the modal approximation by comparing it to numerical solutions based on the finite element method.
This paper extends the high-order entropy stable (ES) adaptive moving mesh finite difference schemes developed in [14] to the two- and three-dimensional (multi-component) compressible Euler equations with the stiffened equation of state. The two-point entropy conservative (EC) flux is first constructed in the curvilinear coordinates. The high-order semi-discrete EC schemes are given with the aid of the two-point EC flux and the high-order discretization of the geometric conservation laws, and then the high-order semi-discrete ES schemes satisfying the entropy inequality are derived by adding the high-order dissipation term based on the multi-resolution weighted essentially non-oscillatory (WENO) reconstruction for the scaled entropy variables to the EC schemes. The explicit strong-stability-preserving Runge-Kutta methods are used for the time discretization and the mesh points are adaptively redistributed by iteratively solving the mesh redistribution equations with an appropriately chosen monitor function. Several 2D and 3D numerical tests are conducted on the parallel computer system with the MPI programming to validate the accuracy and the ability to capture effectively the localized structures of the proposed schemes.
In this paper, we consider the scattering of a plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in three dimensions. Based on the Helmholtz decomposition, the elastic scattering problem is reduced to a coupled boundary value problem for the Helmholtz and Maxwell equations. A novel system of boundary integral equations is formulated and a spectral boundary integral method is developed for the coupled boundary value problem. Numerical experiments are presented to demonstrate the superior performance of the proposed method.
Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires taking into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature for the history integral is to approximate the fractional kernel with the sum of exponentials, which is equivalent to considering the FDE solution as a sum of solutions to a system of ODEs. One possibility to construct this system is to approximate the Laplace spectrum of the fractional kernel with a rational function. In this paper, we use the adaptive Antoulas--Anderson (AAA) algorithm for the rational approximation of the kernel spectrum which yields only a small number of real valued poles. We propose a numerical scheme based on this idea and study its stability and convergence properties. In addition, we apply the algorithm to a time-fractional Cahn-Hilliard problem.
Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line segment is minimized. In the continuous median line segment problem, a real number $\ell>0$ is given, and the goal is to locate a line segment of length $\ell$ in $\mathbb{R}^d$ such that the sum of the Euclidean distances between $P$ and the line segment is minimized. We show how to compute $(1+\epsilon\Delta)$- and $(1+\epsilon)$-approximations to a discrete median line segment in time $O(n\epsilon^{-2d}\log n)$ and $O(n^2\epsilon^{-d})$, respectively, where $\Delta$ is the spread of line segments spanned by pairs of points. While developing our algorithms, by using the principle of pair decomposition, we derive new data structures that allow us to quickly approximate the sum of the distances from a set of points to a given line segment or point. To our knowledge, our utilization of pair decompositions for solving minsum facility location problems is the first of its kind; it is versatile and easily implementable. We prove that it is impossible to construct a continuous median line segment for $n\geq3$ non-collinear points in the plane by using only ruler and compass. In view of this, we present an $O(n^d\epsilon^{-d})$-time algorithm for approximating a continuous median line segment in $\mathbb{R}^d$ within a factor of $1+\epsilon$. The algorithm is based upon generalizing the point-segment pair decomposition from the discrete to the continuous domain. Last but not least, we give an $(1+\epsilon)$-approximation algorithm, whose time complexity is sub-quadratic in $n$, for solving the constrained median line segment problem in $\mathbb{R}^2$ where an endpoint or the slope of the median line segment is given at input.
In $d$ dimensions, approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$) suffers from the pollution effect if, as $k\to \infty$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter 2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania, Sauter 2013] showed that the $hp$-FEM (where accuracy is increased by decreasing the meshwidth $h$ and increasing the polynomial degree $p$) applied to a variety of constant-coefficient Helmholtz problems does not suffer from the pollution effect. The heart of the proofs of these results is a PDE result splitting the solution of the Helmholtz equation into "high" and "low" frequency components. The main novelty of the present paper is that we prove this splitting for the constant-coefficient Helmholtz equation in full-space (i.e., in $\mathbb{R}^d$) using only integration by parts and elementary properties of the Fourier transform (this is contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses somewhat-involved bounds on Bessel and Hankel functions). We combine this splitting with (i) standard arguments about convergence of the FEM applied to the Helmholtz equation (the so-called "Schatz argument", which we reproduce here) and (ii) polynomial-approximation results (which we quote from the literature without proof) to give a simple proof that the $hp$-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation.
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.
We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.
This paper is concerned with the numerical solution of compressible fluid flow in a fractured porous medium. The fracture represents a fast pathway (i.e., with high permeability) and is modeled as a hypersurface embedded in the porous medium. We aim to develop fast-convergent and accurate global-in-time domain decomposition (DD) methods for such a reduced fracture model, in which smaller time step sizes in the fracture can be coupled with larger time step sizes in the subdomains. Using the pressure continuity equation and the tangential PDEs in the fracture-interface as transmission conditions, three different DD formulations are derived; each method leads to a space-time interface problem which is solved iteratively and globally in time. Efficient preconditioners are designed to accelerate the convergence of the iterative methods while preserving the accuracy in time with nonconforming grids. Numerical results for two-dimensional problems with non-immersed and partially immersed fractures are presented to show the improved performance of the proposed methods.
Solutions of the Helmholtz equation are known to be well approximated by superpositions of propagative plane waves. This observation is the foundation of successful Trefftz methods. However, when too many plane waves are used, the computation of the expansion is known to be numerically unstable. We explain how this effect is due to the presence of exponentially large coefficients in the expansion and can drastically limit the efficiency of the approach. In this work, we show that the Helmholtz solutions on a disk can be exactly represented by a continuous superposition of evanescent plane waves, generalizing the standard Herglotz representation. Here, by evanescent plane waves, we mean exponential plane waves with complex-valued propagation vector, whose absolute value decays exponentially in one direction. In addition, the density in this representation is proved to be uniformly bounded in a suitable weighted Lebesgue norm, hence overcoming the instability observed with propagative plane waves and paving the way for stable discrete expansions. In view of practical implementations, discretization strategies are investigated. We construct suitable finite-dimensional sets of evanescent plane waves using sampling strategies in a parametric domain. Provided one uses sufficient oversampling and regularization, the resulting approximations are shown to be both controllably accurate and numerically stable, as supported by numerical evidence.
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