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Computation of nonclassical shocks using a spacetime discontinuous Galerkin method

K. Jegdic
2005 2005 Richard Tapia Celebration of Diversity in Computing Conference  
We present a numerical study for two systems of conservation laws using a spacetime discontinuous Galerkin (SDG) method with causal spacetime triangulations and the piecewise constant Galerkin basis.  ...  The SDG method is consistent with the weak formulation of conservation laws, and, in the case of strictly hyperbolic systems, also with the Lax entropy condition.  ...  However, the numerical experiments presented in this paper show that the SDG method can be successfully used in approximating solutions to more general systems of conservation laws.  ... 
doi:10.1109/rtcdc.2005.201639 fatcat:ka2xnakb5jeijhqbgusrjcmmbq

Computation of nonclassical shocks using a spacetime discontinuous galerkin method

Katarina Jegdic
2005 Proceedings of the 2005 conference on Diversity in computing - TAPIA '05  
We present a numerical study for two systems of conservation laws using a spacetime discontinuous Galerkin (SDG) method with causal spacetime triangulations and the piecewise constant Galerkin basis.  ...  The SDG method is consistent with the weak formulation of conservation laws, and, in the case of strictly hyperbolic systems, also with the Lax entropy condition.  ...  However, the numerical experiments presented in this paper show that the SDG method can be successfully used in approximating solutions to more general systems of conservation laws.  ... 
doi:10.1145/1095242.1095256 dblp:conf/tapia/Jegdic05 fatcat:y66vfiajgfa4zgcvy2g2m2sprm

Page 3113 of Mathematical Reviews Vol. , Issue 2000e [page]

2000 Mathematical Reviews  
Sherwin, A high order Fou- rier/unstructured discontinuous Galerkin method for hyperbolic conservation laws (875-884); Armen Shirikyan and Leonid Vole- vich, Asymptotic properties of solutions to high-order  ...  Galkin, Global correctness of Cauchy problem for nonlinear conservation laws systems and one example for the gas dynamics (361-367); Isabelle Gallagher, Asymptotics for hyper- bolic equations with a skew-symmetric  ... 

On the Galerkin/Finite-Element Method for the Serre Equations

Dimitrios Mitsotakis, Boaz Ilan, Denys Dutykh
2014 Journal of Scientific Computing  
Acknowledgments 32 References 32 On the Galerkin method for the Serre equations Comparing the cB and Serre systems, Eqs. (1.3a) and (1.1a) are the same.  ...  For this reason, the Serre system is often called fully-nonlinear shallow-water equations.  ...  No stability or convergence results are known for the nonlinear (semi-or fully-) discrete schemes for the Serre equations.  ... 
doi:10.1007/s10915-014-9823-3 fatcat:u65etoxm4vbx7nqhapdallj3iy

On the Galerkin / finite-element method for the Serre equations [article]

Dimitrios Mitsotakis
2013 arXiv   pre-print
A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime.  ...  The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge-Kutta method in time.  ...  No stability or convergence results are known for the nonlinear (semi-or fully-) discrete schemes for the Serre equations.  ... 
arXiv:1306.3321v3 fatcat:6as6zaakffehflyfj34cteseg4

A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system

D. Mitsotakis, C. Synolakis, M. McGuinness
2016 International Journal for Numerical Methods in Fluids  
A new modified Galerkin / Finite Element Method is proposed for the numerical solution of the fully nonlinear shallow water wave equations.  ...  After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions  ...  Acknowledgment The authors were supported by the Marsden Fund administered by the Royal Society of New Zealand.  ... 
doi:10.1002/fld.4293 fatcat:7kj5tetjfnbh5i4lysbudhzlpi

Page 1043 of Mathematical Reviews Vol. , Issue 91B [page]

1991 Mathematical Reviews  
In Part II the method is extended to finite element methods for scalar conservation laws in one space dimension.  ...  The quasimonotone method is a hybridization of a monotone method with a high order accurate method, designed to give stability near discontinuities as well as high order accuracy.  ... 

A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

Mohammad Kazem Sharifian, Georges Kesserwani, Yousef Hassanzadeh
2018 Ocean Modelling  
9 This work extends a robust second-order Runge-Kutta Discontinuous Galerkin (RKDG2) 10 method to solve the fully nonlinear and weakly dispersive flows, within a scope to 11 simultaneously address accuracy  ...  In the context of numerically solving elliptic equations with higher order derivatives, 100 often the so-called Local Discontinuous Galerkin (LDG) method is employed as proposed in 101 Cockburn and Shu  ...  Acknowledgments 650 The authors are grateful for two anonymous reviewers for their insightful reviews, which  ... 
doi:10.1016/j.ocemod.2018.03.006 fatcat:k4cb7myhavgxhidyvuhcisi2jm

Page 2249 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews  
Serre, Slow dynamics of linear waves in nonlinear systems of conservation laws (247-260). A. Jeffrey [Alan Jeffrey] and Y. G.  ...  Jie Shen and Roger Temam, Nonlinear Galerkin method using Legendre polynomials (363-376); Joel Smoller and Blake Temple, Multi-dimensional shock-waves for relativistic fluids (377-391); Hesheng Sun, Nonlinear  ... 

Recent advances in Serre–Green Naghdi modelling for wave transformation, breaking and runup processes

P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche, M. Tissier
2011 European journal of mechanics. B, Fluids  
The ability of the Serre or Green Naghdi (S-GN) equations to reproduce this nonlinear processes is reviewed.  ...  Two high-order methods for solving S-GN equations, based on Finite Volume approaches, are presented.  ...  This work has also been supported by the ANR MathOcean, the ANR MISEEVA and the project ECOS-CONYCIT action C07U01.  ... 
doi:10.1016/j.euromechflu.2011.02.005 fatcat:tiwjov6s5bhhvn5dacg7sii5ne

Page 4920 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
Lin, Characteristic Galerkin methods for hyperbolic prob- lems (430-439); Ingo Miiller and Wolf Weiss, The symmetric hy- perbolic equations of extended thermodynamics (440-448); Jean- Philippe Nicolas,  ...  Zhao [Yan Chun Zhao], A generalized Lagrangian method for solving the Euler equations (336-346).  ... 

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

Saray Busto, Michael Dumbser, Cipriano Escalante, Nicolas Favrie, Sergey Gavrilyuk
2021 Journal of Scientific Computing  
We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation.  ...  The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions.  ...  Discontinuous Galerkin finite element schemes for the solution of nonlinear dispersive Boussinesq-type equations were forwarded in [50, 54, 55] .  ... 
doi:10.1007/s10915-021-01429-8 fatcat:gmojrsmqljfndpb2nppqtnab2a

A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies [article]

Caterina Bassi, Saray Busto Ulloa (2 and 3), Michael Dumbser MOX Modelling and Scientific Computing - Dipartimento di Matematica - Politecnico di Milano, INdAM Istituto Nazionale di Alta Matematica Francesco Severi, Laboratory of Applied Mathematics DICAM - University of Trento)
2020 arXiv   pre-print
The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes.  ...  We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves.  ...  for the Environment and Security (SIES).  ... 
arXiv:2003.14309v1 fatcat:wjkehkqmxfdulk465adktdpmby

Third-order Finite Volume/Finite Element Solution of the Fully Nonlinear Weakly Dispersive Serre Equations [article]

Christopher Zoppou, Jordan Pitt, Stephen G. Roberts
2016 arXiv   pre-print
The reformulated Serre equations involve the water depth and a new quantity as the conserved variables which are evolved using the finite volume method.  ...  The nonlinear weakly dispersive Serre equations contain higher-order dispersive terms. This includes a mixed derivative flux term which is difficult to handle numerically.  ...  Geological Survey for providing the rectangular wave data.  ... 
arXiv:1607.08019v1 fatcat:hfgu2xg3s5habbjjmru5mwifxe

Page 1352 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
Pinder, Single- degree freedom collocation method using Hermite polynomials (489-499); Xijun Yu and Yonghong Wu, A Taylor-Galerkin finite element method for one-dimensional hyperbolic conservation laws  ...  Wheeler, Non conforming methods for transport with nonlinear reaction (421—432); Louis F.  ... 
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