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. Dedicated to our friend Saul Abarbanel on the occasion of his 80th birthday. M. Medvinsky · E. Turkel School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel M. Medvinsky e-mail: [email protected] url: http://www.tau.ac.il/~medvinsk/ E. Turkel e-mail: [email protected] url: http://www.math.tau.ac.il/~turkel/ M. Medvinsky · S. Tsynkov ( ) Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA e-mail: [email protected] url: http://www.math.ncsu.edu/~stsynkov J Sci Comput (2012) 53:150–193 151 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.