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Journal articles on the topic 'Discontinous Galerkin method'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Krasnov, Mikhail Mikhailovich, Marina Eugenievna Ladonkina, and Vladimir Fedorovich Tishkin. "Implementation of the Galerkin discontinous method in the DGM software package." Keldysh Institute Preprints, no. 245 (2018): 1–31. http://dx.doi.org/10.20948/prepr-2018-245.

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2

Dolejší, V., M. Feistauer, and C. Schwab. "On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow." Mathematica Bohemica 127, no. 2 (2002): 163–79. http://dx.doi.org/10.21136/mb.2002.134171.

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3

Cockburn, B. "Discontinuous Galerkin methods." ZAMM 83, no. 11 (November 3, 2003): 731–54. http://dx.doi.org/10.1002/zamm.200310088.

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4

Krasnov, M. M., P. A. Kuchugov, M. E. Ladonkina, and V. F. Tishkin. "Efficient parallel software system for solving Navier-Stokes equations by the discontinuous Galerkin method." Computational Mathematics and Information Technologies 2 (2017): 148–55. http://dx.doi.org/10.23947/2587-8999-2017-2-148-155.

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5

Zhang, Xu-jiu, Yong-sheng Zhu, Ke Yan, and You-yun Zhang. "A Front Tracking Method Based on Runge-Kutta Discontinuous Galerkin Methods." International Journal of Online Engineering (iJOE) 12, no. 12 (December 25, 2016): 67. http://dx.doi.org/10.3991/ijoe.v12i12.6453.

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In this paper, a high-resolution front tracking method was presented for interface tracking simulation with Runge-Kutta discontinuous Galerkin methods. An interface treating method of the discontinuous methods is presented. This method don’t construct the ghost fluid and the flow information on both sides next to the interface is used to solve the interfacial status. The limiter adopted the combination of the shock detection and monotonicity-preserving limiter and level set method is used for tracking the interface. Result shown that the front tracking of the high-order accurate Runge-Kutta discontinuous Galerkin method exhibits very good agreement with exact solution in the interface condition that contain strong shock.

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6

Xu, Liyang, Xinhai Xu, Xiaoguang Ren, Yunrui Guo, Yongquan Feng, and Xuejun Yang. "Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure." Advances in Mechanical Engineering 11, no. 10 (October 2019): 168781401985558. http://dx.doi.org/10.1177/1687814019855586.

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In this work, we deal with high-order solver for incompressible flow based on velocity correction scheme with discontinuous Galerkin discretized velocity and standard continuous approximated pressure. Recently, small time step instabilities have been reported for pure discontinuous Galerkin method, in which both velocity and pressure are discretized by discontinuous Galerkin. It is interesting to examine these instabilities in the context of mixed discontinuous Galerkin–continuous Galerkin method. By means of numerical investigation, we find that the discontinuous Galerkin–continuous Galerkin method shows great stability at the same configuration. The consistent velocity divergence discretization scheme helps to achieve more accurate results at small time step size. Since the equal order discontinuous Galerkin–continuous Galerkin method does not satisfy inf-sup stability requirement, the instability for high Reynolds number flow is investigated. We numerically demonstrate that fine mesh resolution and high polynomial order are required to obtain a robust system. With these conclusions, discontinuous Galerkin–continuous Galerkin method is able to achieve high-order spatial convergence rate and accurately simulate high Reynolds flow. The solver is tested through a series of classical benchmark problems, and efficiency improvement is proved against pure discontinuous Galerkin scheme.

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7

Zhang, Rongpei, Xijun Yu, Jiang Zhu, Abimael F. D. Loula, and Xia Cui. "Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation." Communications in Computational Physics 14, no. 5 (November 2013): 1287–303. http://dx.doi.org/10.4208/cicp.190612.010313a.

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AbstractWeighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

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8

Zienkiewicz, O. C., R. L. Taylor, S. J. Sherwin, and J. Peiró. "On discontinuous Galerkin methods." International Journal for Numerical Methods in Engineering 58, no. 8 (August 6, 2003): 1119–48. http://dx.doi.org/10.1002/nme.884.

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9

Hu, Qingjie, Yinnian He, Tingting Li, and Jing Wen. "A Mixed Discontinuous Galerkin Method for the Helmholtz Equation." Mathematical Problems in Engineering 2020 (May 4, 2020): 1–9. http://dx.doi.org/10.1155/2020/9582583.

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In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.

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10

Gopalakrishnan, J., and G. Kanschat. "A multilevel discontinuous Galerkin method." Numerische Mathematik 95, no. 3 (September 1, 2003): 527–50. http://dx.doi.org/10.1007/s002110200392.

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11

Gopalakrishnan, J., and G. Kanschat. "A multilevel discontinuous Galerkin method." Numerische Mathematik 95, no. 3 (September 1, 2003): 551. http://dx.doi.org/10.1007/s00211-003-0504-7.

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12

Warburton, T. C., I. Lomtev, Y. Du, S. J. Sherwin, and G. E. Karniadakis. "Galerkin and discontinuous Galerkin spectral/hp methods." Computer Methods in Applied Mechanics and Engineering 175, no. 3-4 (July 1999): 343–59. http://dx.doi.org/10.1016/s0045-7825(98)00360-0.

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13

Barrenechea, Gabriel R., Michał Bosy, Victorita Dolean, Frédéric Nataf, and Pierre-Henri Tournier. "Hybrid Discontinuous Galerkin Discretisation and Domain Decomposition Preconditioners for the Stokes Problem." Computational Methods in Applied Mathematics 19, no. 4 (October 1, 2019): 703–22. http://dx.doi.org/10.1515/cmam-2018-0005.

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AbstractSolving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis of the discretisation method is presented, and the results are corroborated by numerical experiments. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

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14

Zhang, Xiao, Xiaoping Xie, and Shiquan Zhang. "An Optimal Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems." Computational Methods in Applied Mathematics 19, no. 4 (October 1, 2019): 849–61. http://dx.doi.org/10.1515/cmam-2018-0007.

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AbstractThe embedded discontinuous Galerkin (EDG) method by Cockburn, Gopalakrishnan and Lazarov [B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal. 47 2009, 2, 1319–1365] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second-order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees {k+1}, {k+1}, k ({k\geq 0}) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.

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15

Liu, Donghuan, and Yinghua Liu. "Applications of Discontinuous Galerkin Finite Element Method in Thermomechanical Coupling Problems with Imperfect Thermal Contact." Mathematical Problems in Engineering 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/861417.

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Thermomechanical coupling problems with imperfect thermal contact are analyzed in the present paper with discontinuous Galerkin finite element method. The imperfect thermal contact condition is characterized by thermal contact resistance. The whole thermomechanical coupling problem is solved alternatively with the thermal subproblem and mechanical subproblem. Thermal contact resistance is introduced directly with the interface numerical flux of the present discontinuous Galerkin finite element method without using interface element as traditional continuous Galerkin finite element method does. Numerical results show the accuracy and feasibility of the present discontinuous Galerkin finite element method in solving thermomechanical coupling problems with imperfect thermal contact.

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16

Wei, Leilei, and Xindong Zhang. "A Computational Study of an Implicit Local Discontinuous Galerkin Method for Time-Fractional Diffusion Equations." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/898217.

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We propose, analyze, and test a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully, we prove that our scheme is unconditionally stable and convergent. Finally, numerical examples are performed to illustrate the effectiveness and the accuracy of the method.

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17

Wei, Leilei, Yinnian He, and Xindong Zhang. "Analysis of an Implicit Fully Discrete Local Discontinuous Galerkin Method for the Time-Fractional Kdv Equation." Advances in Applied Mathematics and Mechanics 7, no. 4 (May 29, 2015): 510–27. http://dx.doi.org/10.4208/aamm.2013.m220.

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AbstractIn this paper, we consider a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Korteweg-de Vries (KdV) equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditionally stable and convergent through analysis. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.

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18

Ren, Zongxiu, Leilei Wei, Yinnian He, and Shaoli Wang. "NUMERICAL ANALYSIS OF AN IMPLICIT FULLY DISCRETE LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE FRACTIONAL ZAKHAROV–KUZNETSOV EQUATION." Mathematical Modelling and Analysis 17, no. 4 (September 1, 2012): 558–70. http://dx.doi.org/10.3846/13926292.2012.708675.

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In this paper we develop and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Zakharov–Kuznetsov equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditional stable and L 2 error estimate for the linear case with the convergence rate through analysis.

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19

Qiu, Meilan, Liquan Mei, and Dewang Li. "Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space Fractional Subdiffusion/Superdiffusion Equations." Advances in Mathematical Physics 2017 (2017): 1–20. http://dx.doi.org/10.1155/2017/4961797.

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We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuous Galerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemes for approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractional derivative by using the backward Euler difference for the derivative parameter (0<α<1) or second-order central difference method for (1<α<2), combined with local discontinuous Galerkin method to approximate the spatial derivative which is defined by a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method, we show the concrete analysis for the stability and the convergence under theL2norm of the LDG schemes. Several numerical experiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. The numerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerful for solving fractional partial differential equations.

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20

Xie, Ziqing, Bo Wang, and Zhimin Zhang. "Space-Time Discontinuous Galerkin Method for Maxwell’s Equations." Communications in Computational Physics 14, no. 4 (October 2013): 916–39. http://dx.doi.org/10.4208/cicp.230412.271212a.

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AbstractA fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate is established under the L2-norm when polynomials of degree atmost r and k are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

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21

Zhang, Rongpei, Xijun Yu, Mingjun Li, and Zhen Wang. "A semi-implicit integration factor discontinuous Galerkin method for the non-linear heat equation." Thermal Science 23, no. 3 Part A (2019): 1623–28. http://dx.doi.org/10.2298/tsci180921232z.

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In this paper, a new discontinuous Galerkin method is employed to study the non-linear heat conduction equation with temperature dependent thermal conductivity. We present practical implementation of the new discontinuous Galerkin scheme with weighted flux averages. The second-order implicit integration factor for time discretization method is applied to the semi discrete form. We obtain the L2 stability of the discontinuous Galerkin scheme. Numerical examples show that the error estimates are of second order when linear element approximations are applied. The method is applied to the non-linear heat conduction equations with source term.

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22

Busch, K., M. König, and J. Niegemann. "Discontinuous Galerkin methods in nanophotonics." Laser & Photonics Reviews 5, no. 6 (May 2, 2011): 773–809. http://dx.doi.org/10.1002/lpor.201000045.

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23

Kessler, Manuel. "Viscosity in discontinuous Galerkin methods." PAMM 7, no. 1 (December 2007): 4100039–40. http://dx.doi.org/10.1002/pamm.200700897.

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24

Schnepp, S., and T. Weiland. "Discontinuous Galerkin methods with transienthpadaptation." Radio Science 46, no. 5 (June 8, 2011): n/a. http://dx.doi.org/10.1029/2010rs004639.

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25

Carstensen, C., P. Bringmann, F. Hellwig, and P. Wriggers. "Nonlinear discontinuous Petrov–Galerkin methods." Numerische Mathematik 139, no. 3 (March 6, 2018): 529–61. http://dx.doi.org/10.1007/s00211-018-0947-5.

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26

Göttlich, Simone, and Patrick Schindler. "Discontinuous Galerkin Method for Material Flow Problems." Mathematical Problems in Engineering 2015 (2015): 1–15. http://dx.doi.org/10.1155/2015/341893.

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For the simulation of material flow problems based on two-dimensional hyperbolic partial differential equations different numerical methods can be applied. Compared to the widely used finite volume schemes we present an alternative approach, namely, the discontinuous Galerkin method, and explain how this method works within this framework. An extended numerical study is carried out comparing the finite volume and the discontinuous Galerkin approach concerning the quality of solutions.

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27

Lai, Wencong, and Abdul A. Khan. "Time stepping in discontinuous Galerkin method." Journal of Hydrodynamics 25, no. 3 (June 2013): 321–29. http://dx.doi.org/10.1016/s1001-6058(11)60370-4.

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28

Freund, Jouni. "The space-continuous–discontinuous Galerkin method." Computer Methods in Applied Mechanics and Engineering 190, no. 26-27 (March 2001): 3461–73. http://dx.doi.org/10.1016/s0045-7825(00)00279-6.

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29

Richter, Gerard R. "The discontinuous Galerkin method with diffusion." Mathematics of Computation 58, no. 198 (May 1, 1992): 631. http://dx.doi.org/10.1090/s0025-5718-1992-1122076-2.

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30

Liu, Xiaodong, Nathaniel R. Morgan, and Donald E. Burton. "A Lagrangian discontinuous Galerkin hydrodynamic method." Computers & Fluids 163 (February 2018): 68–85. http://dx.doi.org/10.1016/j.compfluid.2017.12.007.

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31

Wang, Qiuliang, and Jinru Chen. "An Unfitted Discontinuous Galerkin Method for Elliptic Interface Problems." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/241890.

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An unfitted discontinuous Galerkin method is proposed for the elliptic interface problems. Based on a variant of the local discontinuous Galerkin method, we obtain the optimal convergence for the exact solutionuin the energy norm and its fluxpin theL2norm. These results are the same as those in the case of elliptic problems without interface. Finally, some numerical experiments are presented to verify our theoretical results.

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32

Abdulle, Assyr. "Multiscale method based on discontinuous Galerkin methods for homogenization problems." Comptes Rendus Mathematique 346, no. 1-2 (January 2008): 97–102. http://dx.doi.org/10.1016/j.crma.2007.11.029.

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33

Zhou, Zhenhua, and Haijun Wu. "Convergence and Quasi-Optimality of an Adaptive Multi-Penalty Discontinuous Galerkin Method." Numerical Mathematics: Theory, Methods and Applications 9, no. 1 (February 2016): 51–86. http://dx.doi.org/10.4208/nmtma.2015.m1412.

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AbstractAn adaptive multi-penalty discontinuous Galerkin method (AMPDG) for the diffusion problem is considered. Convergence and quasi-optimality of the AMPDG are proved. Compared with the analyses for the adaptive finite element method or the adaptive interior penalty discontinuous Galerkin method, extra works are done to overcome the difficulties caused by the additional penalty terms.

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34

JAFFRE, J., C. JOHNSON, and A. SZEPESSY. "CONVERGENCE OF THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC CONSERVATION LAWS." Mathematical Models and Methods in Applied Sciences 05, no. 03 (May 1995): 367–86. http://dx.doi.org/10.1142/s021820259500022x.

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We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q≥0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.

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35

BÖSING, PAULO R., ALEXANDRE L. MADUREIRA, and IGOR MOZOLEVSKI. "A NEW INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD FOR THE REISSNER–MINDLIN MODEL." Mathematical Models and Methods in Applied Sciences 20, no. 08 (August 2010): 1343–61. http://dx.doi.org/10.1142/s0218202510004623.

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We introduce an interior penalty discontinuous Galerkin finite element method for the Reissner–Mindlin plate model that, as the plate's half-thickness ϵ tends to zero, recovers a hp interior penalty discontinuous Galerkin finite element methods for biharmonic equation. Our method does not introduce shear as an extra unknown, and does not need reduced integration techniques. We develop the a priori error analysis of these methods and prove error bounds that are optimal in h and uniform in ϵ. Numerical tests, that confirm our predictions, are provided.

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36

Walkington, Noel J. "Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions." SIAM Journal on Numerical Analysis 42, no. 5 (January 2005): 1801–17. http://dx.doi.org/10.1137/s0036142902412233.

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37

Liang, X., A. Q. M. Khaliq, and Y. Xing. "Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations." Communications in Computational Physics 17, no. 2 (January 23, 2015): 510–41. http://dx.doi.org/10.4208/cicp.060414.190914a.

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AbstractThis paper studies a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

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Wrana, B. "A Discontinuous Galerkin Finite Element Method for Dynamic of Fully Saturated Soil / Rzwiazanie Zadania Dynamiki Całkowicie Nawodnionego Gruntu Przy Zastosowaniu Mes Z Nieciagłym Sformułowaniem Galerkina W Czasie." Archives of Civil Engineering 57, no. 1 (March 1, 2011): 119–34. http://dx.doi.org/10.2478/v.10169-011-0009-1.

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Abstract The fully coupled, porous solid-fluid dynamic field equations with u-p formulation are used in this paper to simulate pore fluid and solid skeleton responses. The present formulation uses physical damping, which dissipates energy by velocity proportional damping. The proposed damping model takes into account the interaction of pore fluid and solid skeleton. The paper focuses on formulation and implementation of Time Discontinuous Galerkin (TDG) methods for soil dynamics in the case of fully saturated soil. This method uses both the displacements and velocities as basic unknowns and approximates them through piecewise linear functions which are continuous in space and discontinuous in time. This leads to stable and third-order accurate solution algorithms for ordinary differential equations. Numerical results using the time-discontinuous Galerkin FEM are compared with results using a conventional central difference, Houbolt, Wilson θ, HHT- α, and Newmark methods. This comparison reveals that the time-discontinuous Galerkin FEM is more stable and more accurate than these traditional methods.

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39

Kuang, Yangyu, Kailiang Wu, and Huazhong Tang. "Runge-Kutta Discontinuous Local Evolution Galerkin Methods for the Shallow Water Equations on the Cubed-Sphere Grid." Numerical Mathematics: Theory, Methods and Applications 10, no. 2 (May 2017): 373–419. http://dx.doi.org/10.4208/nmtma.2017.s09.

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AbstractThe paper develops high order accurate Runge-Kutta discontinuous local evolution Galerkin (RKDLEG) methods on the cubed-sphere grid for the shallow water equations (SWEs). Instead of using the dimensional splitting method or solving one-dimensional Riemann problem in the direction normal to the cell interface, the RKDLEG methods are built on genuinely multi-dimensional approximate local evolution operator of the locally linearized SWEs on a sphere by considering all bicharacteristic directions. Several numerical experiments are conducted to demonstrate the accuracy and performance of our RKDLEG methods, in comparison to the Runge-Kutta discontinuous Galerkin method with Godunov's flux etc.

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40

Liu, Yingjie, Chi-Wang Shu, Eitan Tadmor, and Mengping Zhang. "L2stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods." ESAIM: Mathematical Modelling and Numerical Analysis 42, no. 4 (May 27, 2008): 593–607. http://dx.doi.org/10.1051/m2an:2008018.

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41

Calvo, Lucas, Diana De Padova, Michele Mossa, and Paulo Rosman. "Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup." Computation 9, no. 4 (April 14, 2021): 47. http://dx.doi.org/10.3390/computation9040047.

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This paper presents a new depth-integrated non-hydrostatic finite element model for simulating wave propagation, breaking and runup using a combination of discontinuous and continuous Galerkin methods. The formulation decomposes the depth-integrated non-hydrostatic equations into hydrostatic and non-hydrostatic parts. The hydrostatic part is solved with a discontinuous Galerkin finite element method to allow the simulation of discontinuous flows, wave breaking and runup. The non-hydrostatic part led to a Poisson type equation, where the non-hydrostatic pressure is solved using a continuous Galerkin method to allow the modeling of wave propagation and transformation. The model uses linear quadrilateral finite elements for horizontal velocities, water surface elevations and non-hydrostatic pressures approximations. A new slope limiter for quadrilateral elements is developed. The model is verified and validated by a series of analytical solutions and laboratory experiments.

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42

Archibald, Rick, Katherine J. Evans, John Drake, and James B. White. "Multiwavelet Discontinuous Galerkin-Accelerated Exact Linear Part (ELP) Method for the Shallow-Water Equations on the Cubed Sphere." Monthly Weather Review 139, no. 2 (February 1, 2011): 457–73. http://dx.doi.org/10.1175/2010mwr3271.1.

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Abstract In this paper a new approach is presented to increase the time-step size for an explicit discontinuous Galerkin numerical method. The attributes of this approach are demonstrated on standard tests for the shallow-water equations on the sphere. The addition of multiwavelets to the discontinuous Galerkin method, which has the benefit of being scalable, flexible, and conservative, provides a hierarchical scale structure that can be exploited to improve computational efficiency in both the spatial and temporal dimensions. This paper explains how combining a multiwavelet discontinuous Galerkin method with exact-linear-part time evolution schemes, which can remain stable for implicit-sized time steps, can help increase the time-step size for shallow-water equations on the sphere.

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43

Dazel, Olivier, and Gwenael Gabard. "Discontinuous Galerkin Methods for poroelastic materials." Journal of the Acoustical Society of America 133, no. 5 (May 2013): 3242. http://dx.doi.org/10.1121/1.4805189.

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Kreuzer, Christian, and Emmanuil H. Georgoulis. "Convergence of adaptive discontinuous Galerkin methods." Mathematics of Computation 87, no. 314 (February 26, 2018): 2611–40. http://dx.doi.org/10.1090/mcom/3318.

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Bringmann, P., C. Carstensen, D. Gallistl, F. Hellwig, D. Peterseim, S. Puttkammer, H. Rabus, and J. Storn. "Towards adaptive discontinuous Petrov-Galerkin methods." PAMM 16, no. 1 (October 2016): 741–42. http://dx.doi.org/10.1002/pamm.201610359.

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Richter, Andreas, Jörg Stiller, and Roger Grundmann. "Discontinuous Galerkin Methods for Aeroacoustical Investigations." PAMM 8, no. 1 (December 2008): 10697–98. http://dx.doi.org/10.1002/pamm.200810697.

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Sherwin, S. J., R. M. Kirby, J. Peiró, R. L. Taylor, and O. C. Zienkiewicz. "On 2D elliptic discontinuous Galerkin methods." International Journal for Numerical Methods in Engineering 65, no. 5 (2005): 752–84. http://dx.doi.org/10.1002/nme.1466.

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Donoghue, Geoff, and Masayuki Yano. "Spatio-stochastic adaptive discontinuous Galerkin methods." Computer Methods in Applied Mechanics and Engineering 374 (February 2021): 113570. http://dx.doi.org/10.1016/j.cma.2020.113570.

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Kretzschmar, Fritz, Sascha M. Schnepp, Igor Tsukerman, and Thomas Weiland. "Discontinuous Galerkin methods with Trefftz approximations." Journal of Computational and Applied Mathematics 270 (November 2014): 211–22. http://dx.doi.org/10.1016/j.cam.2014.01.033.

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McLachlan, Robert I., and Ari Stern. "Multisymplecticity of Hybridizable Discontinuous Galerkin Methods." Foundations of Computational Mathematics 20, no. 1 (April 22, 2019): 35–69. http://dx.doi.org/10.1007/s10208-019-09415-1.

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