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We develop a hybrid spatial discretization for the wave equation in secondorder form, based on high-order accurate finite difference methods anddiscontinuous Galerkin methods. The hybridization combines computationalefficiency of finite difference methods on Cartesian grids and geometricalflexibility of discontinuous Galerkin methods on unstructured meshes. The twospatial discretizations are coupled by a penalty technique at the interfacesuch that the overall semidiscretization satisfies a discrete energy estimateto ensure stability. In addition, optimal convergence is obtained in the sensethat when combining a fourth order finite difference method with adiscontinuous Galerkin method using third order local polynomials, the overallconvergence rate is fourth order. Furthermore, we use a novel approach toderive an error estimate for the semidiscretization by combining the energymethod and the normal mode analysis for a corresponding one dimensional modelproblem. The stability and accuracy analysis are verified in numericalexperiments.

A finite difference - discontinuous Galerkin method for the wave equation in second order form