Date of Award
Master of Science
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.
Samuel Quincy Van Fleet
Van Fleet, Samuel Quincy, "A Lax-Wendroff discontinuous Galerkin scheme for linear hyperbolic systems" (2020). Graduate Theses and Dissertations. 18240.