A Conservation Constrained Runge-Kutta Discontinuous Galerkin Method with the Improved CFL Condition for Conservation Laws

We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [6, 5, 4, 3] 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 polynomial approximation. The hierarchical reconstruction [12, 24] 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.

[1]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[2]  Chi-Wang Shu,et al.  Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction , 2007, SIAM J. Numer. Anal..

[3]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[4]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[5]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[6]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[7]  Eitan Tadmor,et al.  Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes , 2006 .

[8]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[9]  Zhiliang Xu,et al.  Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells , 2009, J. Comput. Phys..

[10]  Bram Van Leer,et al.  Discontinuous Galerkin for Diffusion , 2005 .

[11]  Chi-Wang Shu.  TVB uniformly high-order schemes for conservation laws , 1987 .

[12]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[13]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case , 2005 .

[14]  Rainald Löhner,et al.  A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids , 2007, J. Comput. Phys..

[15]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[16]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[17]  ShuChi-Wang,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes, II , 1989 .

[18]  G. Lin,et al.  SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS , 2009 .

[19]  B. Vanleer,et al.  Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection , 1977 .

[20]  Timothy C. Warburton,et al.  Taming the CFL Number for Discontinuous Galerkin Methods on Structured Meshes , 2008, SIAM J. Numer. Anal..

[21]  Michael Dumbser,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[22]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[23]  Chi-Wang Shu,et al.  L2 Stability Analysis of the Central Discontinuous Galerkin Method and a Comparison between the Central and Regular Discontinuous Galerkin Methods , 2008 .