A Runge-Kutta Discontinuous Galerkin Method with Conservation Constraints to Improve CFL Condition for Solving Conservation Laws

We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [7, 6, 5, 4] for conservation Laws. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. We use this new formulation to solve conservation laws on one-dimensional grids with piecewise cubic polynomial approximation as well as on two-dimensional unstructured grids with piecewise quadratic and cubic polynomial approximation. The hierarchical reconstruction [13, 25] is applied as a limiter to eliminate spurious oscillations. Numerical computations for scalar and systems of nonlinear hyperbolic conservation laws are performed. We find that: 1) this new formulation improves the CFL number over the original RKDG formulation and thus reduces the overall complexity; 2) the new formulation improves the robustness of the DG scheme with the current limiting strategy and improves the resolution of the numerical solutions in multi-dimensions.