Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers? (English) Zbl 1488.65562
Summary: This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series of solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 65F05 | Direct numerical methods for linear systems and matrix inversion |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 78A50 | Antennas, waveguides in optics and electromagnetic theory |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
Keywords:
Helmholtz equation; separation of variables; high wave number; pollution free finite difference scheme; large cavity problem; rectangular waveguideReferences:
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