Development of Residual Distribution Schemes for theDiscontinuous GalerkinMethod: The ScalarCase with Linear Elements

In this paper, we reformulate the piecewise linear discontinuous Galerkin (DG) method for solving two dimensional steady state scalar conservation laws in the framework of residual distribution (RD) schemes. This allows us to propose a new class of nonlinear stabilization that does not destroy the formal accuracy of the schemes. Numerical results are shown to demonstrate the behavior of this approach.

[1]  Rémi Abgrall,et al.  Toward the ultimate conservative scheme: following the quest , 2001 .

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

[3]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics , 1986 .

[4]  Rémi Abgrall,et al.  High Order Fluctuation Schemes on Triangular Meshes , 2003, J. Sci. Comput..

[5]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems , 1986 .

[6]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[7]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[8]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[9]  Herman Deconinck,et al.  Residual distribution for general time-dependent conservation laws , 2005 .

[10]  Rémi Abgrall,et al.  Towards very high‐order accurate schemes for unsteady convection problems on unstructured meshes , 2005 .

[11]  Chi-Wang Shu,et al.  High order residual distribution conservative finite difference WENO schemes for convection-diffusion steady state problems on non-smooth meshes , 2007, J. Comput. Phys..

[12]  Rémi Abgrall,et al.  Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems , 2003 .