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Higher order A-stable schemes for the wave equation using a recursive convolution approach

In several recent works \cite{Causley2013a}, \cite{Causley2013}, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL$^T$) formulation combined with alternating direction implicit (ADI) schemes. In this work, we present several important modifications to our work, and thus obtain a family of wave solvers which are unconditionally stable, accurate of order 2P, and require $O(P^d N)$ operations per time step, where $N$ is the number of spatial points, and $d$ the number of spatial dimensions. We obtain these schemes by including higher derivatives of the solution, rather than increasing the number of time levels. The novel aspect of our approach is that the higher derivatives are constructed using successive applications of the convolution operator. We develop these schemes in one spatial dimension, and then extend the results to higher dimensions, by reformulating the ADI scheme to include recursive convolution. Thus, we retain a fast, unconditionally stable scheme, which does not suffer from the large dispersion errors characteristic to the ADI method. We demonstrate the utility of the method by applying it to a host of wave propagation problems. This method holds great promise for developing higher order, parallelizable algorithms for solving hyperbolic PDEs, and can also be extended to parabolic PDEs.