A Lax-Wendroff discontinuous Galerkin scheme for linear hyperbolic systems

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2020-01-01
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Van Fleet, Samuel
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James Rossmanith
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We develop in this work a Lax-Wendroff discontinuous Galerkin (LxW-DG) scheme for solving linear systems of hyperbolic partial differential equations (PDEs). The proposed scheme is a variant of the standard LxW-DG scheme from the literature. The process by which the standard LxW-DG 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; and finally, (4) replace the boundary terms by appropriate numerical fluxes. The key innovation in the newly proposed scheme 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 standard LxW-DG approach. We develop this newly proposed approach in both one and two spatial dimensions and compare the regions of stability to the standard LxW-DG scheme. We show that compared to the standard LxW-DG scheme, the modified scheme has a larger region of stability and has improved accuracy. We demonstrate the properties of this new scheme by applying it to several numerical test cases.

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Sat Aug 01 00:00:00 UTC 2020