A dynamic p-adaptive Discontinuous Galerkin method for viscous flow with shocks

Abstract A dynamic p -adaptive Discontinuous Galerkin method for compressible flows is proposed. The key element is a sensor, which measures the local regularity of the solution. This sensor is designed to preserve the compactness of the method and is easy to implement. In regions where the solution is quasi-uniform and in the vicinity of shocks the degree of the polynomial basis is decreased, while a high-degree basis is used in regions of smooth fluctuations of the flow. Numerical tests carried out in 1D and on 2D unstructured meshes prove that the convergence rate of the hybrid method is equal to the one of the highest degree polynomial basis, but with a significant CPU cost saving.

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

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

[3]  Bernardo Cockburn,et al.  Discontinuous Galerkin Methods for Convection-Dominated Problems , 1999 .

[4]  George Em Karniadakis,et al.  A triangular spectral element method; applications to the incompressible Navier-Stokes equations , 1995 .

[5]  P. Sagaut,et al.  Simulation of a viscous compressible flow past a circular cylinder with high-order discontinuous Galerkin methods , 2002 .

[6]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[7]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[8]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[9]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[10]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[11]  Chi-Wang Shu,et al.  High Order ENO and WENO Schemes for Computational Fluid Dynamics , 1999 .

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

[13]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives , 2002, J. Sci. Comput..

[14]  Pierre Sagaut,et al.  A problem-independent limiter for high-order Runge—Kutta discontinuous Galerkin methods , 2001 .

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

[16]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

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

[18]  Timothy J. Barth,et al.  High-order methods for computational physics , 1999 .