The Runge-Kutta discontinuous Galerkin method with compact stencils for hyperbolic conservation laws. (English) Zbl 07837063
Summary: In this paper, we develop a new type of Runge-Kutta (RK) discontinuous Galerkin (DG) method for solving hyperbolic conservation laws. Compared with the original RKDG method, the new method features improved compactness and allows simple boundary treatment. The key idea is to hybridize two different spatial operators in an explicit RK scheme, utilizing local projected derivatives for inner RK stages and the usual DG spatial discretization for the final stage only. Limiters are applied only at the final stage for the control of spurious oscillations. We also explore the connections between our method and Lax-Wendroff DG schemes and ADER-DG schemes. Numerical examples are given to confirm that the new RKDG method is as accurate as the original RKDG method, while being more compact, for problems including two-dimensional Euler equations for compressible gas dynamics.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
Keywords:
discontinuous Galerkin method; Runge-Kutta method; stencil size; high-order Numerical method; convergence; hyperbolic conservation lawsSoftware:
ADER-DGReferences:
| [1] | Bassi, F., Crivellini, A., Rebay, S., and Savini, M., Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-\( \omega\) turbulence model equations, Comput. & Fluids, 34 (2005), pp. 507-540. · Zbl 1138.76043 |
| [2] | Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., and Savini, M., A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, in Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, , Antwerpen, Belgium, Technologische Instituut, Antwerpen, Belgium1997, pp. 99-109. |
| [3] | Bona, J. L., Chen, H., Karakashian, O. A., and Xing, Y., Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Math. Comp., 82 (2013), pp. 1401-1432. · Zbl 1276.65058 |
| [4] | Carpenter, M. H., Gottlieb, D., Abarbanel, S., and Don, W.-S., The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A study of the boundary error, SIAM J. Sci. Comput., 16 (1995), pp. 1241-1252. · Zbl 0839.65098 |
| [5] | Cheng, Y. and Shu, C.-W., A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp., 77 (2008), pp. 699-730. · Zbl 1141.65075 |
| [6] | Cockburn, B., Hou, S., and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp., 54 (1990), pp. 545-581. · Zbl 0695.65066 |
| [7] | Cockburn, B., Lin, S.-Y., and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys., 84 (1989), pp. 90-113. · Zbl 0677.65093 |
| [8] | Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp., 52 (1989), pp. 411-435. · Zbl 0662.65083 |
| [9] | Cockburn, B. and Shu, C.-W., The Runge-Kutta local projection \({P}^1\)-discontinuous-Galerkin finite element method for scalar conservation laws, ESAIM Math. Model. Numer. Anal., 25 (1991), pp. 337-361. · Zbl 0732.65094 |
| [10] | Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199-224. · Zbl 0920.65059 |
| [11] | Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), pp. 173-261. · Zbl 1065.76135 |
| [12] | Dumbser, M., Balsara, D. S., Toro, E. F., and Munz, C.-D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227 (2008), pp. 8209-8253. · Zbl 1147.65075 |
| [13] | Dumbser, M., Enaux, C., and Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227 (2008), pp. 3971-4001. · Zbl 1142.65070 |
| [14] | Dumbser, M. and Munz, C.-D., Building blocks for arbitrary high order discontinuous Galerkin schemes, J. Sci. Comput., 27 (2006), pp. 215-230. · Zbl 1115.65100 |
| [15] | Gaburro, E., Öffner, P., Ricchiuto, M., and Torlo, D., High order entropy preserving ADER-DG schemes, Appl. Math. Comput., 440 (2023), 127644. · Zbl 1511.65109 |
| [16] | Gottlieb, S., Ketcheson, D. I., and Shu, C.-W., Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations, World Scientific, Hackensack, NJ, 2011. · Zbl 1241.65064 |
| [17] | Gottlieb, S., Shu, C.-W., and Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89-112. · Zbl 0967.65098 |
| [18] | Grant, Z. J., Perturbed Runge-Kutta methods for mixed precision applications, J. Sci. Comput., 92 (2022), 6. · Zbl 1491.65064 |
| [19] | Guo, W., Qiu, J.-M., and Qiu, J., A new Lax-Wendroff discontinuous Galerkin method with superconvergence, J. Sci. Comput., 65 (2015), pp. 299-326. · Zbl 1333.65110 |
| [20] | Luo, H., Luo, L., Nourgaliev, R., Mousseau, V. A., and Dinh, N., A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, J. Comput. Phys., 229 (2010), pp. 6961-6978. · Zbl 1425.35138 |
| [21] | Peraire, J. and Persson, P.-O., The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30 (2008), pp. 1806-1824. · Zbl 1167.65436 |
| [22] | Qiu, J., A numerical comparison of the Lax-Wendroff discontinuous Galerkin method based on different numerical fluxes, J. Sci. Comput., 30 (2007), pp. 345-367. · Zbl 1176.76080 |
| [23] | Qiu, J., Dumbser, M., and Shu, C.-W., The discontinuous Galerkin method with Lax-Wendroff type time discretizations, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 4528-4543. · Zbl 1093.76038 |
| [24] | Rannabauer, L., Dumbser, M., and Bader, M., ADER-DG with a-posteriori finite-volume limiting to simulate tsunamis in a parallel adaptive mesh refinement framework, Comput. & Fluids, 173 (2018), pp. 299-306. · Zbl 1410.86006 |
| [25] | Reed, W. H. and Hill, T., Triangular Mesh Methods for the Neutron Transport Equation, Technical report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, NM, 1973. |
| [26] | Shi, C. and Shu, C.-W., On local conservation of numerical methods for conservation laws, Comput. & Fluids, 169 (2018), pp. 3-9. · Zbl 1410.65327 |
| [27] | Shu, C.-W. and Osher, S., Efficient implementation of essentially nonoscillatory shock-capturing schemes, II, J. Comput. Phys., 83 (1989), pp. 32-78. · Zbl 0674.65061 |
| [28] | Sun, Z. and Shu, C.-W., Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws, ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 1063-1087. · Zbl 1373.65063 |
| [29] | Titarev, V. A. and Toro, E. F., ADER: Arbitrary high order Godunov approach, J. Sci. Comput., 17 (2002), pp. 609-618. · Zbl 1024.76028 |
| [30] | Toro, E. F., Millington, R., and Nejad, L., Towards very high order Godunov schemes, in Godunov Methods: Theory and Applications, Springer, New York, 2001, pp. 907-940. · Zbl 0989.65094 |
| [31] | van Leer, B., Lo, M., and van Raalte, M., A discontinuous Galerkin method for diffusion based on recovery, in 18th AIAA Computational Fluid Dynamics Conference, , 2007, 2007-4083. |
| [32] | Van Leer, B. and Nomura, S., Discontinuous Galerkin for diffusion, in 17th AIAA Computational Fluid Dynamics Conference, , AIAA, Reston, VA, 2005, 2005-5108. |
| [33] | Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115-173. · Zbl 0573.76057 |
| [34] | Yan, J. and Shu, C.-W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40 (2002), pp. 769-791. · Zbl 1021.65050 |
| [35] | Zhang, Q., Third order explicit Runge-Kutta discontinuous Galerkin method for linear conservation law with inflow boundary condition, J. Sci. Comput., 46 (2011), pp. 294-313. · Zbl 1258.76115 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
