Stability analysis and error estimate of the explicit single-step time-marching discontinuous Galerkin methods with stage-dependent numerical flux parameters for a linear hyperbolic equation in one dimension. (English) Zbl 07902990
Summary: In this paper, we present the \(\mathrm{L}^2\)-norm stability analysis and error estimate for the explicit single-step time-marching discontinuous Galerkin (DG) methods with stage-dependent numerical flux parameters, when solving a linear constant-coefficient hyperbolic equation in one dimension. Two well-known examples of this method include the Runge-Kutta DG method with the downwind treatment for the negative time marching coefficients, as well as the Lax-Wendroff DG method with arbitrary numerical flux parameters to deal with the auxiliary variables. The stability analysis framework is an extension and an application of the matrix transferring process based on the temporal differences of stage solutions, and a new concept, named as the averaged numerical flux parameter, is proposed to reveal the essential upwind mechanism in the fully discrete status. Distinguished from the traditional analysis, we have to present a novel way to obtain the optimal error estimate in both space and time. The main tool is a series of space-time approximation functions for a given spatial function, which preserve the local structure of the fully discrete schemes and the balance of exact evolution under the control of the partial differential equation. Finally some numerical experiments are given to validate the theoretical results proposed in this paper.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35E15 | Initial value problems for PDEs and systems of PDEs with constant coefficients |
Keywords:
discontinuous Galerkin method; explicit single step time marching; stage-dependent numerical flux parameters; hyperbolic equation; stability analysis; error estimateReferences:
| [1] | Ai, J.; Xu, Y.; Shu, CW; Zhang, Q., \({\rm L}^2\) error estimate to smooth solutions of high order Runge-Kutta discontinuous Galerkin method for scalar nonlinear conservation laws with and without sonic points, SIAM J. Numer. Anal., 60, 4, 1741-1773, 2022 · Zbl 1501.65062 · doi:10.1137/21M1435495 |
| [2] | Chavent, G.; Cockburn, B., The local projection \(P^0P^1\)-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér., 23, 4, 565-592, 1989 · Zbl 0715.65079 · doi:10.1051/m2an/1989230405651 |
| [3] | Cheng, Y.; Meng, X.; Zhang, Q., Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp., 86, 305, 1233-1267, 2017 · Zbl 1359.65196 · doi:10.1090/mcom/3141 |
| [4] | Ciarlet, P.G.: The finite element method for elliptic problems. In: Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Co., New York (1978) · Zbl 0383.65058 |
| [5] | Cockburn, B.; Hou, S.; Shu, CW, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comput., 54, 190, 545-581, 1990 · Zbl 0695.65066 · doi:10.2307/2008501 |
| [6] | Cockburn, B.; Lin, SY; Shu, CW, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems, J. Comput. Phys., 84, 1, 90-113, 1989 · Zbl 0677.65093 · doi:10.1016/0021-9991(89)90183-6 |
| [7] | Cockburn, B.; Shu, CW, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comput., 52, 186, 411-435, 1989 · Zbl 0662.65083 · doi:10.2307/2008474 |
| [8] | Cockburn, B.; Shu, CW, The Runge-Kutta local projection \(P^1\)-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér., 25, 3, 337-361, 1991 · Zbl 0732.65094 · doi:10.1051/m2an/1991250303371 |
| [9] | Cockburn, B.; Shu, CW, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys., 141, 2, 199-224, 1998 · Zbl 0920.65059 · doi:10.1006/jcph.1998.5892 |
| [10] | Cockburn, B.; Shu, CW, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 3, 173-261, 2001 · Zbl 1065.76135 · doi:10.1023/A:1012873910884 |
| [11] | Gottlieb, S.; Ruuth, SJ, Optimal strong-stability-preserving time-stepping schemes with fast downwind spatial discretizations, J. Sci. Comput., 27, 1-3, 289-303, 2006 · Zbl 1115.65092 · doi:10.1007/s10915-005-9054-8 |
| [12] | Gottlieb, S.; Shu, CW, Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 221, 73-85, 1998 · Zbl 0897.65058 · doi:10.1090/S0025-5718-98-00913-2 |
| [13] | Guo, W.; Qiu, J.; Qiu, J., A new Lax-Wendroff discontinuous Galerkin method with superconvergence, J. Sci. Comput., 65, 1, 299-326, 2015 · Zbl 1333.65110 · doi:10.1007/s10915-014-9968-0 |
| [14] | Liu, Y.; Shu, CW; Zhang, M., Sub-optimal convergence of discontinuous Galerkin methods with central fluxes for linear hyperbolic equations with even degree polynomial approximations, J. Comput. Math., 39, 4, 518-537, 2021 · Zbl 1499.65507 · doi:10.4208/jcm.2002-m2019-0305 |
| [15] | Qiu, J., Zhang, Q.: Stability, error estimate and limiters of discontinuous Galerkin methods. In: Handbook of Numerical Methods for Hyperbolic Problems, Handbook of Numerical Analysis, vol. 17, pp. 147-171. Elsevier, Amsterdam (2016). doi:10.1016/bs.hna.2016.06.001 · Zbl 1352.65001 |
| [16] | Ruuth, SJ, Global optimization of explicit strong-stability-preserving Runge-Kutta methods, Math. Comput., 75, 253, 183-207, 2006 · Zbl 1080.65088 · doi:10.1090/S0025-5718-05-01772-2 |
| [17] | Ruuth, SJ; Spiteri, RJ, Two barriers on strong-stability-preserving time discretization methods, J. Sci. Comput., 17, 1-4, 211-220, 2002 · Zbl 1003.65107 · doi:10.1023/A:1015156832269 |
| [18] | Ruuth, SJ; Spiteri, RJ, High-order strong-stability-preserving Runge-Kutta methods with downwind-biased spatial discretizations, SIAM J. Numer. Anal., 42, 3, 974-996, 2004 · Zbl 1089.65069 · doi:10.1137/S0036142902419284 |
| [19] | Shu, CW, Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9, 6, 1073-1084, 1988 · Zbl 0662.65081 · doi:10.1137/0909073 |
| [20] | Shu, C.W.: Discontinuous Galerkin methods: general approach and stability. In: Numerical Solutions of Partial Differential Equations, Adv. Courses Math. CRM Barcelona, pp. 149-201. Birkhäuser, Basel (2009) |
| [21] | Shu, C.W.: Discontinuous Galerkin methods for time-dependent convection dominated problems: basics, recent developments and comparison with other methods. In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computer Science Engineering, vol. 114, pp. 369-397. Springer, New York (2016) |
| [22] | Shu, CW; Osher, S., Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77, 2, 439-471, 1988 · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5 |
| [23] | Sun, Z.; Shu, CW, Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws, ESAIM Math. Model. Numer. Anal., 51, 3, 1063-1087, 2017 · Zbl 1373.65063 · doi:10.1051/m2an/2016049 |
| [24] | Sun, Z.; Shu, CW, Strong stability of explicit Runge-Kutta time discretizations, SIAM J. Numer. Anal., 57, 3, 1158-1182, 2019 · Zbl 1422.65224 · doi:10.1137/18M122892X |
| [25] | Van Loan, CF, The ubiquitous Kronecker product, J. Comput. Appl. Math., 123, 1-2, 85-100, 2000 · Zbl 0966.65039 · doi:10.1016/S0377-0427(00)00393-9 |
| [26] | Xu, Y.; Meng, X.; Shu, CW; Zhang, Q., Superconvergence analysis of the Runge-Kutta discontinuous Galerkin methods for a linear hyperbolic equation, J. Sci. Comput., 84, 23, 2020 · Zbl 1465.65105 · doi:10.1007/s10915-020-01274-1 |
| [27] | Xu, Y.; Shu, CW; Zhang, Q., Error estimate of the fourth-order Runge-Kutta discontinuous Galerkin methods for linear hyperbolic equations, SIAM J. Numer. Anal., 58, 5, 2885-2914, 2020 · Zbl 1452.65256 · doi:10.1137/19M1280077 |
| [28] | Xu, Y.; Zhang, Q., Superconvergence analysis of the Runge-Kutta discontinuous Galerkin method with upwind-biased numerical flux for two dimensional linear hyperbolic equation, Commun. Appl. Math. Comput., 4, 319-352, 2022 · Zbl 1499.65541 · doi:10.1007/s42967-020-00116-z |
| [29] | Xu, Y.; Zhang, Q.; Shu, CW; Wang, H., The \(\text{L}^2\)-norm stability analysis of Runge-Kutta discontinuous Galerkin methods for linear hyperbolic equations, SIAM J. Numer. Anal., 57, 4, 1574-1601, 2019 · Zbl 1422.65272 · doi:10.1137/18M1230700 |
| [30] | Xu, Y.; Zhao, D.; Zhang, Q., Local error estimates for Runge-Kutta discontinuous Galerkin methods with upwind-biased numerical fluxes for a linear hyperbolic equation in one-dimension with discontinuous initial data, J. Sci. Comput., 91, 11, 2022 · Zbl 1491.65103 · doi:10.1007/s10915-022-01793-z |
| [31] | Zhang, Q.; Shu, CW, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal., 42, 2, 641-666, 2004 · Zbl 1078.65080 · doi:10.1137/S0036142902404182 |
| [32] | Zhang, Q.; Shu, CW, Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal., 48, 3, 1038-1063, 2010 · Zbl 1217.65178 · doi:10.1137/090771363 |
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