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Title: Genuinely multi-dimensional, maximum Taylor discontinuous Galerkin schemes for solving linear hyperbolic systems of conservation laws
Abstract: In this work we develop the maximum Taylor discontinuous Galerkin (MTDG) method for solving linear systems of hyperbolic partial differential equations (PDEs). The proposed method is a variant of the Lax-Wendroff discontinuous Galerkin (LxW-DG) method from the literature. The process by which the Lax-Wendroff DG method is obtained can be summarized as follows: 1. Compute a truncated Taylor series in time that relates the solution that is being sought to the known solution at the previous time-step. 2. Replace all the temporal derivatives in this Taylor expansion by spatial derivatives by repeatedly invoking the underlying PDE. 3. Multiply this expansion by appropriate test functions, integrate over a finite element, and perform a single integration-by-parts that places a derivative on the test functions as well as introducing boundary terms. 4. Replace the boundary terms by appropriate numerical fluxes. The key innovation in the newly proposed method is that we replace the single integration-by-parts step by an approach that moves all spatial derivatives onto the test functions; this process introduces many new terms that are not present in the Lax-Wendroff DG approach. The regions of stability various MTDG methods are compared to the LxW-DG stability regions. It is shown that compared to the Lax-Wendroff DG method, the maximum Taylor DG method has a larger region of stability and has improved accuracy. These properties are demonstrated by applying MTDG to several numerical test cases.