Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

Abstract

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

A high order finite difference method for the elastic wave equation in bounded domains with nonconforming interfaces

“…Over the years, the SBP operators G(γ) and G(γ) have been independently used to solve wave propagation problems, see [16,17] for G(γ) and [2,6] for G(γ). In [22], an algorithm was developed to convert G(γ) by Sjögreen and Petersson to an SBP operator G(γ) that does not use ghost points. By using the combination of G(γ) and G(γ), an improved coupling procedure at nonconforming grid interfaces was developed for the wave equation in [22], and extended to the elastic wave equations in [23].…”

“…In [22], an algorithm was developed to convert G(γ) by Sjögreen and Petersson to an SBP operator G(γ) that does not use ghost points. By using the combination of G(γ) and G(γ), an improved coupling procedure at nonconforming grid interfaces was developed for the wave equation in [22], and extended to the elastic wave equations in [23]. In both [22] and [23], periodic boundary conditions are used in the spatial directions tangential to the interface, which allows standard centered interpolation stencils to be used on all hanging nodes.…”

“…For the SBP operators with ghost points, we use the fourth order version by Sjögreen and Petersson [18]. Since the SBP operator with ghost points based on the sixth order interior stencil has not been published, we use the technique from [22] and convert Mattsson's operator to a new SBP operator with ghost points. At the nonconforming interface, we use the OP interpolation to couple mesh blocks.…”

Non-stiff boundary and interface penalties for narrow-stencil finite difference approximations of the Laplacian on curvilinear multiblock grids

“…Furthermore, the new discretization is presented for d spatial dimensions and applies to the generalized Laplace operator ∇ · b( x)∇. This paper only considers weak enforcement of boundary and interface conditions using SATs, but it is possible to combine SBP operators with strong enforcement by, for example, injection [4] or ghost points [20,31]. In fact, for certain classes of boundary and interface conditions, the ghost point technique yields a significantly smaller spectral radius than SATs [31].…”

“…This paper only considers weak enforcement of boundary and interface conditions using SATs, but it is possible to combine SBP operators with strong enforcement by, for example, injection [4] or ghost points [20,31]. In fact, for certain classes of boundary and interface conditions, the ghost point technique yields a significantly smaller spectral radius than SATs [31]. We stress that the purpose of this paper is not to find the finite difference discretization of the Laplacian with the smallest possible spectral radius for a given set of boundary conditions.…”