title: A New Runge-Kutta Discontinuous Galerkin Method with Conservation Constraint to Improve CFL Condition for Solving Conservation Laws.
abstract: We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [9, 8, 7, 6] for solving conservation Laws with increased CFL numbers. 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. Numerical computations for solving one-dimensional and two-dimensional scalar and systems of nonlinear hyperbolic conservation laws are performed with approximate solutions represented by piecewise quadratic and cubic polynomials, respectively. The hierarchical reconstruction [17, 33] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. From both numerical experiments and the analytic estimate of the CFL number of the newly formulated method, we find that: 1) this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2) the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method.
authors.affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556.
authors.forename: Zhiliang
authors.initials: Z
authors.lastname: Xu
authors.affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332.
authors.forename: Xu-Yan
authors.initials: XY
authors.lastname: Chen
authors.affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332.
authors.forename: Yingjie
authors.initials: Y
authors.lastname: Liu