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We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (μ(x)u x ) x . This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (μ(x)u x ) x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.

Difference approximations are derived for the second order wave equation in one and two space dimensions, without first writing it as a first order system. Both the Dirichlet and the Neumann problems are treated for the one-dimensional case. Relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary. General two-dimensional domains are considered for the Dirichlet problem where the domain is embedded in a Cartesian grid and the boundary conditions are approximated by interpolation. A stable conservative scheme is derived where the time step is determined only by the interior discretization formula. Discretization cells cut by the boundary are treated implicitly, but the resulting scheme becomes explicit because the implicit dependence only is pointwise. Numerical examples are provided to verify the stability and accuracy of the proposed method.

Stability theory and numerical experiments are presented for a finite difference method that directly discretizes the Neumann problem for the second order wave equation. Complex geometries are discretized using a Cartesian embedded boundary technique. Both second and third order accurate approximations of the boundary conditions are presented. Away from the boundary, the basic second order method can be corrected to achieve fourth order spatial accuracy. To integrate in time, we present both a second order and a fourth order accurate explicit method. The stability of the method is ensured by adding a small fourth order dissipation operator, locally modified near the boundary to allow its application at all grid points inside the computational domain. Numerical experiments demonstrate the accuracy and long-time stability of the proposed method.

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