Hyperbolic model for Helmholtz equation with impedance boundary conditions
This provides a novel computational approach for solving high-wavenumber Helmholtz problems, which are notoriously difficult for standard numerical methods.
The authors reformulate the Helmholtz equation with impedance boundary conditions as a hyperbolic system that reaches steady state in finite time, enabling high accuracy for large wavenumbers with few spatial and temporal nodes.
Solution of Helmholtz equation with impedance boundary condition on finite interval is equivalently reformulated as steady state of initial boundary value problem for first order hyperbolic system of partial differential equations. Particularly interesting property of the proposed hyperbolic model is that steady state is achieved in finite time. For large wavenumber the numerically challenging task for Helmholtz equation is achieving high accuracy with small number of nodal points. We successfully solved this problem by means of using well balanced scheme approach. Numerical tests demonstrate excellent computational potential of the proposed method: high accuracy is achieved for large wavenumber with small number of nodal points in space and time.