Preprint 2013-006
A new Runge–Kutta discontinuous Galerkin method with conservation constraint to improve CFL condition for solving conservation laws
Zhiliang Xu, Xu-Yan Chen and Yingjie Liu
Abstract: We present a new formulation of the Runge–Kutta discontinuous Galerkin (RKDG) method [7, 6, 5, 4] for solving 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 one-dimensional and two-dimensional conservation laws with piecewise quadratic and cubic polynomial approximation, respectively. The hierarchical reconstruction [13, 25] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. 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 computational cost; 2) the new formulation improves the robustness of the DG scheme with the current limiting strategy and improves the resolution of the numerical solutions of shock wave problems in multi-dimensions.
[4] B. Cockburn and C.-W. Shu. The TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws V: multidimensional systems. J. Comput. Phys., 141:199–224, 1998.
[5] B. Cockburn, S. Hou and C.-W. Shu. The TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comp., 54:545–581, 1990.
[6] B. Cockburn, S.-Y. Lin and C.-W. Shu. TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys., 52:411–435, 1989.
[7] B. Cockburn and C.-W. Shu. TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comp., 52:411– 435, 1989.
[13] Y.-J. Liu, C.-W. Shu, E. Tadmor and M.-P. Zhang. Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction. SIAM J. Numer. Anal., 45:2442–2467, 2007.
[25] Z.-L. Xu and Y.-J. Liu and C.-W. Shu. Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys., 228:2194–2212, 2009.