Finite-difference algorithm with local time-space grid refinement for simulation of waves

Abstract

This paper presents a new approach to a local time-space grid refinement for a staggered-grid finitedifference simulation of waves. The approach is based on approximation of a wave equation at the interface where two grids are coupled. As no interpolation or projection techniques are used, the finite-difference scheme preserves second order of convergence. We have proved that this approach is low-reflecting, the artificial reflections are about 10 −4 of an incident wave. We have also shown that if a successive refinement is applied, i.e. temporal and spatial steps are refined at different interfaces, this approach is stable. Mathematics Subject Classifications (2010) 86A15· 86-08 · 35L40 · 65M06 · 65M50 · 65M55 V. Lisitsa (B) · V. Tcheverda

Local time–space mesh refinement for simulation of elastic wave propagation in multi-scale media

“…To overcome the stability issues associated with the use of discontinuous grids for the numerical simulation, we apply successive mesh refinement [56], in which the steps with respect to time and space are refined at different interfaces. This means that a transition zone with the fine grid T F in time and the coarse spatial grid Ω C is introduced into consideration, as presented in Fig.…”

“…To update the solution at the interface x 3 = J 0 h 3 , let us follow the embedded stencils technique proposed in [56]. Let us treat the following two cases separately:…”

“…To update the velocity vector component, the embedded stencil technique is applied to the second order formulation of the elastic wave equation as was done in [56]. This equation can be derived from (1) excluding the stress tensor components:…”

“…In fact, this approximation can be derived from scheme (8), excluding the components of the stress tensor, in the same manner as for the differential equation. It is easy to prove that all the properties of the staggered grid approximation of the first and second order formulation of the elastic wave equation coincide, see Appendix A in [56]. Subsequently, the embedded stencil technique can be applied to approximate the derivatives with respect to time:…”

“…This means that these approaches are not suitable for the simulation of scattered waves, which have amplitudes of approximately 10 −2 that of an incident wave. Recently we have proposed an approach to the local time-space mesh refinement [56] in which the refinement of temporal and spatial steps is performed at different interfaces, which stabilizes the algorithms. The refinement of the temporal steps is performed using the embedded stencil technique and ensures a low level of reflections from the artificial interface between two grids.…”