A Fourth Order Cut Finite Element (cutFEM) Method for the Wave Equation using the Modifed Equation Approach

A symmetric bilinear form corresponding to the biharmonic operator is combined with a corresponding bilinear form for the Laplacian to create a temporally 4th order corrected leap-frog scheme for the wave equation. The boundary conditions are imposed weakly, which allows for handling an unaligned domain with a stabilised cut-element methodology. The approach requires only one mass matrix solve per time-step. We develop fully discrete cutHermite and cutDG methodologies using cubic basis functions and Cartesian meshes. In the DG case additional symmetric interior penalties are added to impose sufficient continuity, while the higher continuity of the Hermite finite elements suffices. The schemes are shown to be stable under CFL conditions similar to the corresponding un-cut and un-corrected cases, and to yield 4th order accurate solutions. We also discuss how the biharmonic weak form can be used on its own to solve for example a biharmonic equation.