A new family of fourth-order locally one-dimensional schemes for the three-dimensional wave equation

Abstract

In this paper, we present a new family of locally one-dimensional (LOD) schemes with fourth-order accuracy in both space and time for the three-dimensional (3D) acoustic wave equation. It is well-known that high order explicit schemes offer efficient time steppings but with restrictive CFL conditions; implicit discretization gives unconditional stable schemes but with inefficient time steppings. Our LOD schemes can be seen as a compromise between explicit and implicit schemes, in the way that our schemes have more relax CFL conditions compared with traditional explicit schemes and only require solutions of tridiagonal systems of linear algebraic equations, which is more efficient than implicit schemes. Moreover, the new scheme is four-layer in time and three-layer in space, so that boundary conditions can be imposed in the classical way. The computations of the initial conditions for the three intermediate time layers are explicitly constructed. Furthermore, the stability condition is derived and the restriction for the time step is given explicitly. Finally, numerical examples are completed to show the performance of our new method.