A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions

D. S. Britt, S. V. Tsynkov, E. Turkel
2013 SIAM Journal on Scientific Computing  
We describe a high-order accurate methodology for the numerical simulation of time-harmonic waves governed by the Helmholtz equation. Our approach combines compact finite difference schemes that provide an inexpensive venue toward high-order accuracy with the method of difference potentials developed by Ryaben'kii. The latter can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. The method of difference
more » ... als can accommodate non-conforming boundaries on regular structured grids with no loss of accuracy due to staircasing. It introduces a universal framework for treating boundary conditions of any type. A significant advantage of this method is that altering the boundary condition does not require any major changes to the algorithm. In this paper, we address various types of boundary conditions using the method of difference potentials. We demonstrate the resulting numerical capabilities by solving a range of non-standard boundary value problems for the Helmholtz equation. These include problems with variable-coefficient Robin boundary conditions (including discontinuous coefficients) and problems with mixed (Dirichlet/Neumann) boundary conditions. In all of our simulations, we used a Cartesian grid with a circular boundary curve. For those test cases where the overall solution was smooth, our methodology has consistently demonstrated the design fourth-order rate of grid convergence, whereas, when the regularity of the solution was not sufficient, the convergence slowed down, as expected.
doi:10.1137/120902689 fatcat:3b6ijemctzcmnl3jv3uy3unmpe