Analysis and entropy stability of the line-based discontinuous Galerkin method
For computational scientists solving hyperbolic conservation laws, this method offers a more efficient alternative to standard DG while maintaining nonlinear entropy stability, though it is an incremental improvement over existing entropy-stable DG approaches.
The authors developed a line-based discontinuous Galerkin method for hyperbolic conservation laws that guarantees discrete entropy stability via flux differencing, and demonstrated it is significantly less computationally expensive than standard DG methods on test cases including Burgers' and Euler equations in 1D-3D.
We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions.