Exact finite difference schemes for solving Helmholtz equation at any wavenumber.

*(English)*Zbl 1272.65083The authors derive a new finite difference scheme for the Helmholtz equation. The novelty of this method relies on the fact that it satisfies the Helmholtz equation and the radiation boundary conditions exactly. Consequently, the method can be applied also for high frequencies without any condition on the small mesh size. The method is efficient since the underlying matrix has the same sparsity structure as that obtained from the standard finite difference schemes. Numerical experiments confirm that for one-dimensional problems exact solutions are obtained for any wave number. On the other hand, for two-dimensional problems good estimation of the angle is required and it is realized by the least-squares algorithm.

Reviewer: Mária Lukáčová (Mainz)

##### 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 |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |