Michael Medvinsky

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The method of difference potentials was originally proposed by Ryaben’kii and can be interpreted as a generalized discrete version of the method of Calderon’s operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve.Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils—one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation.We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

Local absorbing boundary conditions for elliptical shaped boundaries

We compare several local absorbing boundary conditions for solving the Helmholtz equation, by a finite difference or finite element method, exterior to a general scatterer. These boundary conditions are imposed on an artificial elliptical or prolate spheroid outer surface. In order to compare the computational solution with an analytical solution, we consider, as an example, scattering about an ellipse. We solve the Helmholtz equation with both finite differences and finite elements. We also introduce a new boundary condition for an ellipse based on a modal expansion.

On surface radiation conditions for an ellipse

We compare several On Surface Radiation Boundary Conditions in two dimensions, for solving the Helmholtz equation exterior to an ellipse. We also introduce a new boundary condition for an ellipse based on a modal expansion in Mathieu functions. We compare the OSRC to a finite difference method.

High order numerical simulation of the transmission and scattering of waves using the method of difference potentials

The method of difference potentials generalizes the method of Calderon’s operators from PDEs to arbitrary difference equations and systems. It offers several key advantages, such as the capability of handling boundaries/interfaces that are not aligned with the discretization grid, variable coefficients, and nonstandard boundary conditions. In doing so, the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid. Previously, we have applied the method of difference potentials to solving several variable coefficient interior Helmholtz problems with fourth and sixth order accuracy. We have employed compact finite difference schemes as a core discretization methodology. Those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself. Numerical experiments corroborate the high order accuracy of our method for variable coefficients, regular grids, and non-conforming boundaries. In the current paper, we extend the previously developed methodology to exterior problems. We present a complete theoretical analysis of the algorithm, as well as the results of a series of numerical simulations. Specifically, we study the scattering of time-harmonic waves about smooth shapes, subject to various boundary conditions. We also solve the transmission/scattering problems, in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside. In all the cases, our methodology guarantees high order accuracy for regular grids and non-conforming boundaries and interfaces.

On the Solution of the Elliptic Interface Problems by Difference Potentials Method

Designing numerical methods with high-order accuracy for problems in irregular domains and/or with interfaces is crucial for the accurate solution of many problems with physical and biological applications. The major challenge here is to design an efficient and accurate numerical method that can capture certain properties of analytical solutions in different domains/subdomains while handling arbitrary geometries and complex structures of the domains. Moreover, in general, any standard method (finite-difference, finite-element, etc.) will fail to produce accurate solutions to interface problems due to discontinuities in the model’s parameters/solutions. In this work, we consider Difference Potentials Method (DPM) as an efficient and accurate solver for the variable coefficient elliptic interface problems.

Direct implementation of high order BGT artificial boundary conditions

Abstract Local artificial boundary conditions (ABCs) for the numerical simulation of waves have been successfully used for decades (most notably, the boundary conditions due to Engquist & Majda, Bayliss, Gunzburger & Turkel, and Higdon). The basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution. The larger the number of terms canceled, the higher the order of the boundary condition and, in turn, the smaller the reflection error due to truncation of the original unbounded domain by an artificial outer boundary. In practice, however, the use of local ABCs has been limited to low orders (first and second), because higher order boundary conditions involve higher order derivatives of the solution, which may harm well-posedness and cause numerical instabilities. They are also difficult to implement especially in finite elements. A prominent exception is the development of local high order ABCs based on auxiliary variables. In the current paper, we implement high order Bayliss–Turkel ABCs directly — with no auxiliary variables yet no discrete approximation of the constituent high order derivatives either. Instead, we represent the solution at the boundary as an expansion with respect to a continuous basis. For the spherical artificial boundary, the basis consists of eigenfunctions of the Beltrami operator (spherical harmonics), which enable replacing the high order derivatives in the ABCs with powers of the corresponding eigenvalues. The continuous representation at the boundary is coupled to higher order compact finite differences inside the domain by the method of difference potentials (MDP). It maintains high order accuracy even when the boundary is not aligned with the discretization grid.