Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

doi:10.1007/s42967-021-00149-y

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (3): 855-879.doi: 10.1007/s42967-021-00149-y

• ORIGINAL PAPER • 上一篇下一篇

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

Lu Zhang¹, Daniel Appelö², Thomas Hagstrom³

1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA;
2. Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA;
3. Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA

收稿日期:2020-11-23 修回日期:2021-04-04 出版日期:2022-09-20 发布日期:2022-07-04
通讯作者: Thomas Hagstrom,E-mail:thagstrom@smu.edu;Lu Zhang,E-mail:lz2784@columbia.edu;Daniel Appelö,E-mail:appeloda@msu.edu E-mail:thagstrom@smu.edu;appeloda@msu.edu;appeloda@msu.edu
基金资助:
This work was supported by NSF Grants DMS-1913076 and DMS-2012296 and completed, while the third author was in residence at the Institute for Computational and Experimental Mathematics. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

Lu Zhang¹, Daniel Appelö², Thomas Hagstrom³

1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA;
2. Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA;
3. Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA

Received:2020-11-23 Revised:2021-04-04 Online:2022-09-20 Published:2022-07-04
Contact: Thomas Hagstrom,E-mail:thagstrom@smu.edu;Lu Zhang,E-mail:lz2784@columbia.edu;Daniel Appelö,E-mail:appeloda@msu.edu E-mail:thagstrom@smu.edu;appeloda@msu.edu;appeloda@msu.edu
Supported by:
This work was supported by NSF Grants DMS-1913076 and DMS-2012296 and completed, while the third author was in residence at the Institute for Computational and Experimental Mathematics. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

摘要/Abstract

摘要： We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

关键词: Discontinuous Galerkin, Galerkin difference, Simultaneous diagonalization

Abstract: We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

Key words: Discontinuous Galerkin, Galerkin difference, Simultaneous diagonalization

中图分类号:

65M60
65M06

Lu Zhang, Daniel Appelö, Thomas Hagstrom. Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 855-879.

参考文献

[1] Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)
[2] Ainsworth, M., Monk, P., Muniz, W.: Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27, 5–40 (2006)
[3] Appelö, D., Hagstrom, T.: A new discontinuous Galerkin formulation for wave equations in second order form. SIAM J. Numer. Anal. 53(6), 2705–2726 (2015)
[4] Appelö, D., Hagstrom, T.: An energy-based discontinuous Galerkin discretization of the elastic wave equation in second order form. Comput. Methods Appl. Mech. Eng. 338, 362–391 (2018)
[5] Appelö, D., Hagstrom, T., Wang, Q., Zhang, L.: An energy-based discontinuous Galerkin method for semilinear wave equations. J. Comput. Phys. 418, 109608 (2020)
[6] Banks, J.W., Buckner, B.B., Hagstrom, T., Juhnke, K.: Discontinuous Galerkin Galerkin differences for the wave equation in second-order form. SIAM J. Sci. Comput. 43(2), A1497–A1526 (2021)
[7] Banks, J.W., Hagstrom, T.: On Galerkin difference methods. J. Comput. Phys. 313, 310–327 (2016)
[8] Banks, J.W., Henshaw, W.D.: Upwind schemes for the wave equation in second-order form. J. Comput. Phys. 231(17), 5854–5889 (2012). . Banks, J.W., Henshaw, W.D.: Upwind schemes for the wave equation in second-order form. J. Comput. Phys. 231(17), 5854–5889 (2012). https://doi.org/10.1016/j.jcp.2012.05.012. http://www.sciencedirect.com/science/article/pii/S0021999112002367
[9] Chan, J., Hewett, R.J., Warburton, T.: Weight-adjusted discontinuous Galerkin methods: curvilinear meshes. SIAM J. Sci. Comput. 39(6), A2395–A2421 (2017)
[10] Chan, J., Hewett, R.J., Warburton, T.: Weight-adjusted discontinuous Galerkin methods: wave propagation in heterogeneous media. SIAM J. Sci. Comput. 39(6), A2935–A2961 (2017)
[11] Chou, C.S., Shu, C.-W., Xing, Y.: Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media. J. Comput. Phys. 272, 88–107 (2014). . Chou, C.S., Shu, C.-W., Xing, Y.: Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media. J. Comput. Phys. 272, 88–107 (2014). https://doi.org/10.1016/j.jcp.2014.04.009. http://www.sciencedirect.com/science/article/pii/S0021999114002721
[12] Grote, M.J., Schneebeli, A., Schötzau, D.: Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006). http://www.jstor.org/stable/40232901
[13] Hagstrom, T., Banks, J.W., Buckner, B.B., Juhnke, K.: Discontinuous Galerkin difference methods for symmetric hyperbolic systems. J. Sci. Comput. 81, 1509–1526 (2019)
[14] Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Science & Business Media, Berlin (2007)
[15] Hu, F.Q., Hussaini, M., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151(2), 921–946 (1999)
[16] Jones, M., Plassmann, P.: Algorithm 740: Fortran subroutines to compute improved incomplete Cholesky factorizations. ACM Trans. Math. Soft. (TOMS) 21, 5–17 (1995)
[17] Kozdon, J., Wilcox, L., Hagstrom, T., Banks, J.: Robust approaches to handling complex geometries with Galerkin difference methods. J. Comput. Phys. 392, 483–510 (2019)
[18] Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numer. Math. 6(1), 185–199 (1964)
[19] Mattsson, K.: Summation by parts operators for finite difference approximations of second-derivatives with variable coefficient. J. Sci. Comput. 51, 650–682 (2012)
[20] Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004)
[21] Moura, R.C., Sherwin, S., Peiró, J.: Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods. J. Comput. Phys. 298, 695–710 (2015)
[22] Riviere, B., Wheeler, M.: Discontinuous finite element methods for acoustic and elastic wave problems. Part i: semidiscrete error estimates. Contemp. Math. 329, 271–282 (2003)
[23] Schmitz, P.G., Ying, L.: A fast nested dissection solver for Cartesian 3D elliptic problems using hierarchical matrices. J. Comput. Phys. 258, 227–245 (2014)
[24] Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)
[25] Zhang, L., Hagstrom, T., Appelö, D.: An energy-based discontinuous Galerkin method for the wave equation with advection. SIAM J. Numer. Anal. 57(5), 2469–2492 (2019)

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Xiaozhou Li. How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 759-782.
[2]	Lorenzo Botti, Daniele A. Di Pietro. p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 783-822.
[3]	Ohannes A. Karakashian, Michael M. Wise. A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 823-854.
[4]	Poorvi Shukla, J. J. W. van der Vegt. A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 904-944.
[5]	Dinshaw S. Balsara, Roger Käppeli. Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 945-985.
[6]	Lukáš Vacek, Václav Kučera. Discontinuous Galerkin Method for Macroscopic Traffic Flow Models on Networks[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 986-1010.
[7]	Jiawei Sun, Shusen Xie, Yulong Xing. Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 381-416.
[8]	Jennifer K. Ryan. Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 417-436.
[9]	Mahboub Baccouch. Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 437-476.
[10]	Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang. Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 477-499.
[11]	Xue Hong, Yinhua Xia. Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 530-562.
[12]	Xiaobing Feng, Thomas Lewis, Aaron Rapp. Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 563-596.
[13]	Andreas Dedner, Tristan Pryer. Discontinuous Galerkin Methods for a Class of Nonvariational Problems[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 634-656.
[14]	Andreas Dedner, Robert Klöfkorn. Extendible and Efcient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 657-696.
[15]	Will Pazner, Tzanio Kolev. Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refnement[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 697-727.