Novel adaptive finite volume method on unstructured meshes for time-domain wave scattering and diffraction

A new adaptive finite volume method is proposed for the simulation of the wave problems in the time domain. The transient wave equations are discretized in time and space. A vertex-centered finite volume method is constructed with both cell-centered and edge-midpoint of each control volume. We then propose a mesh adaptation procedure based on energy-norm error-estimates, which significantly increases the efficiency of the method. The proposed approach is accurate in capturing the details of the scattered and diffracted waves with highly refined elements following the evolution of the wave patterns. This is a critical feature of the approach as waves can propagate in vast domains and we successfully refine the mesh only where needed. Unlike many other methods for evolution problems in which the differential operator is solved after each error estimation, the proposed approach allows for multiple adaptations of meshes within a single error estimation. This nested adaptive finite volume method requires only treatment of conformity in meshes for multiple adaptations which is dealt with using a Newest-Vertex-Bisection algorithm. Comparisons with the finite element and with reference solutions are considered for progressive radial waves, waves reflection and a wave scattering and diffraction around barriers. The proposed approach results are more efficient and highly accurate, hence, the significant potential when applied to the time-domain simulation of wave problems.