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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. For scalar PDEs, the proposed method is also equivalent to the semi-Lagrangian discontinuous (SLDG) Galerkin 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. It is also shown that the maximum Taylor discontinuous Galerkin method is equivalent to the semi-Lagrangian discontinuous Galerkin method for scalar problems. This newly proposed approach is developed in both one and two spatial dimensions; the regions of stability varoius MTDG methods are compared to the LxWDG 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.
Finally, we apply the maximum Taylor DG method to the problem of radiatiative transfer. We consider the so-called gray equation, which is a single-energy model of photon or neutron transport. This time-dependent equation is defined on a 5-dimensional phase space that includes 3 position variables and 2 velocity directions. In this work we reduce this high-dimensional PDE to a system of linear PDEs via the so-called PN (i.e., spherical harmonics) ansatz. In particular, in this work the MTDG method is combined with a Strang operator splitting approach to handle the collision terms. Solutions from the two-dimensional line-source test problem are shown for various values of N.
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