A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws
Shuang Z. Tu

Abstract

This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element/Solution Element (CE/SE) method and the discontinuous Galerkin (DG) method. The core idea is to construct a staggered spacetime mesh through alternate cell-centered CEs and vertex-centered CEs within each time step. Inside each SE, the solution is approximated using high-order spacetime DG basis polynomials. The spacetime flux conservation is enforced inside each CE using the DG concept. The unknowns are stored at both vertices and cell centroids of the spatial mesh. However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Thanks to the staggered spacetime formulation, there are no left and right states for the solution at the spacetime interface. Instead, the solution available to evaluate the flux is continuous across the interface. Therefore, no (approximate) Riemann solvers are needed to provide a unique numerical flux. The current method can be used to solve arbitrary conservation laws including the compressible Euler equations, shallow water equations and magnetohydrodynamics (MHD) equations without the need of any form of Riemann solvers. A set of benchmark problems of various conservation laws are presented to demonstrate the accuracy of the method.

Share and Cite:

Tu, S. (2015) A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws. American Journal of Computational Mathematics, 5, 55-74. doi: 10.4236/ajcm.2015.52004.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Roe, P. (1981) Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes. Journal of Computational Physics, 43, 357-372. http://dx.doi.org/10.1016/0021-9991(81)90128-5 [2] Tu, S. (2009) A High Order Space-Time Riemann-Solver-Free Method for Solving Compressible Euler Equations. Proceedings of the 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, 5-8 January 2009, AIAA Paper 2009-1335. http://dx.doi.org/10.2514/6.2009-1335 [3] Tu, S. (2009) A Solution Limiting Procedure for an Arbitrarily High Order Space-Time Method. Proceedings of the 19th AIAA Computational Fluid Dynamics Conference, San Antonio, 22-25 June 2009, AIAA Paper 2009-3983. [4] Tu, S., Skelton, G. and Pang, Q. (2011) A Compact high Order Space-Time Method for Conservation Laws. Communications in Computational Physics, 9, 441-480. [5] Tu, S. and Tian, Z. (2010) Preliminary Implementation of a High Order Space-Time Method on Overset Cartesian/ Quadrilateral Grids. Proceedings of the 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, 4-7 January 2010, AIAA Paper 2010-0544. http://dx.doi.org/10.2514/6.2010-544 [6] Tu, S., Pang, Q. and Xiang, H. (2011) Wave Computation Using a High Order Space-Time Riemann Solver Free Method. Proceedings of the 17th AIAA/CEAS Aeroacoustics Conference, Portland, 5-8 June 2011, AIAA Paper 2011- 2846. [7] Tu, S., Skelton, G. and Pang, Q. (2012) Extension of the High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme to Solve Time Dependent Diffusion Equations. Communications in Computational Physics, 11, 1503-1524. http://dx.doi.org/10.4208/cicp.050810.090611a [8] Tu, S., Pang, Q. and Xiang, H. (2012) Solving the Shallow Water Equations Using the High Order Space-Time Discontinuous Galerkin Cell-Vertex Scheme. Proceedings of the 50th AIAA Aerospace Science Meetings, Nashville, 9-12 January 2012, AIAA Paper 2012-0307. [9] Tu, S., Pang, Q. and Xiang, H. (2012) Solving the Level Set Equation Using the High Order Space-Time Discontinuous Galerkin Cell-Vertex Scheme. Proceedings of the 50th AIAA Aerospace Science Meetings, Nashville, 9-12 January 2012, AIAA Paper 2012-1233. http://dx.doi.org/10.2514/6.2012-1233 [10] Tu, S. and Pang, Q. (2012) Development of the High Order Space-Time Discontinuous Galerkin Cell Vertex Scheme (DG-CVS) for Moving Mesh Problems. Proceedings of the 42nd AIAA Fluid Dynamics Conference, New Orleans, 25- 28 June 2012, AIAA Paper 2012-2835. http://dx.doi.org/10.2514/6.2012-2835 [11] Tu, S., Pang, Q. and Myong, R. (2013) A Riemann-Solver Free Space-Time Discontinuous Galerkin Method for Magnetohydrodynamics. Proceedings of the 44th AIAA Plasmadynamics and Lasers Conference, San Diego, 24-27 June 2013, AIAA Paper 2013-2755. http://dx.doi.org/10.2514/6.2013-2755 [12] Tu, S., Song, H., Ji, L. and Pang, Q. (2015) Further Development of a Riemann-Solver Free Space-Time Discontinuous Galerkin Method for Compressible Magnetohydrodynamics (MHD) Equations. Proceedings of the 53rd AIAA Aerospace Sciences Meeting, Kissimmee, 5-9 January 2015, AIAA Paper 2015-0567. [13] Chang, S.-C. and To, W. (1991) A New Numerical Framework for Solving Conservation Laws: The Method of Space-Time Conservation Element and Solution Element. NASA TM 1991-104495. [14] Cockburn, B. and Shu, C.-W. (2001) Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems. Journal of Scientific Computing, 16, 173-261. http://dx.doi.org/10.1023/A:1012873910884 [15] Dyson, R. (2001) Technique for Very High Order Nonlinear Simulation and Validation. NASA/TM 2001-210985. [16] Dumbser, M., K?serb, M., Titarevb, V.A. and Toro, E.F. (2007) Quadrature-Free Non-Oscillatory Finite Volume Schemes on Unstructured Meshes for Nonlinear Hyperbolic Systems. Journal of Computational Physics, 226, 204-243. http://dx.doi.org/10.1016/j.jcp.2007.04.004 [17] Dasgupta, G. (2003) Integration within Polygonal Finite Elements. Journal of Aerospace Engineering, 16, 9-18. http://dx.doi.org/10.1061/(ASCE)0893-1321(2003)16:1(9) [18] Jiang, G. and Shu, C.-W. (1996) Efficient Implementation of Weighted ENO Schemes. Journal of Computational Phy- sics, 126, 202-228. http://dx.doi.org/10.1006/jcph.1996.0130 [19] Popescu, M., Shyy, W. and Garbey, M. (2005) Finite Volume Treatment of Dispersion-Relation-Preserving and Optimized Prefactored Compact Schemes for Wave Propagation. Journal of Computational Physics, 210, 705-729. http://dx.doi.org/10.1016/j.jcp.2005.05.011 [20] Osher, S. and Sethian, J.A. (1988) Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations. Journal of Computational Physics, 79, 12-49. http://dx.doi.org/10.1016/0021-9991(88)90002-2 [21] Zalesak, S. (1979) Fully Multidimensional Flux-Corrected Transport Algorithms for Fluids. Journal of Computational Physics, 31, 335-362. http://dx.doi.org/10.1016/0021-9991(79)90051-2 [22] Bell, J.B., Colella, P. and Glaz, H.M. (1989) A Second-Order Projection Method for the Incompressible Navier-Stokes Equations. Journal of Computational Physics, 85, 257-283. http://dx.doi.org/10.1016/0021-9991(89)90151-4 [23] Le Veque, R.J. (1996) High Resolution Conservative Algorithms for Advection in Incompressible Flow. SIAM Journal on Numerical Analysis, 33, 627-665. http://dx.doi.org/10.1137/0733033 [24] Tan, W. (1992) Shallow Water Hydrodynamics: Mathematical Theory and Numerical Solution for a Two-Dimensional System of Shallow Water Equations. Elsevier Oceanography Series, Water & Power Press, Beijing. [25] Liang, S.-J. and Hsu, T.-W. (2009) Least-Squares Finite-Element Method for Shallow-Water Equations with Source Terms. Acta Mechanica Sinica, 25, 597-610. http://dx.doi.org/10.1007/s10409-009-0250-x [26] Cueto-Felgueroso, L., Colominas, I., Fe, J., Navarrina, F. and Casteleiro, M. (2005) High-Order Finite Volume Schemes on Unstructured Grids Using Moving Least-Squares Reconstruction. Application to Shallow Water Dynamics. International Journal for Numerical Methods in Engineering, 65, 295-331. http://dx.doi.org/10.1002/nme.1442 [27] Myong, R. and Roe, P. (1997) Shock Waves and Rarefaction Waves in Magnetohydrodynamics. Part 1. A Model System. Journal of Plasma Physics, 58, 485-519. http://dx.doi.org/10.1017/S002237789700593X [28] Myong, R.S. and Roe, P.L. (1998) On Godunov-Type Schemes for Magnetohydrodynamics. Journal of Computational Physics, 147, 545-567. http://dx.doi.org/10.1006/jcph.1998.6101 [29] Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I. and De Zeeuw, D.L. (1999) A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics. Journal of Computational Physics, 154, 284-309. http://dx.doi.org/10.1006/jcph.1999.6299 [30] Brio, M. and Wu, C.C. (1988) An Upwind Differencing Scheme for the Equations of Ideal Magnetohydrodynamics. Journal of Computational Physics, 75, 400-422. http://dx.doi.org/10.1016/0021-9991(88)90120-9 [31] Li, F. and Shu, C.-W. (2005) Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations. Journal of Scientific Computing, 22-23, 413-442. http://dx.doi.org/10.1007/s10915-004-4146-4