# American Institute of Mathematical Sciences

May  2003, 9(3): 559-576. doi: 10.3934/dcds.2003.9.559

## Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system

 1 School of Mathematical Sciences, Capital Normal University, 100037, Beijing, China 2 Arbeitsbereich Mathematik, Technische Universität Hamburg-Harburg, Hamburg, Germany 3 Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

Received  January 2001 Revised  October 2002 Published  February 2003

The subject of this paper is a demonstration of the accuracy and robustness of evolution Galerkin schemes applied to two-dimensional Riemann problems with finitely many constant states. In order to have a test case with known exact solution we consider a linear first order system for the wave equation and test evolution Galerkin methods as well as other commonly used schemes with respect to their accuracy in capturing important structural phenomena of the solution. For the two-dimensional Riemann problems with finitely many constant states some parts of the exact solution are constructed in the following three steps. Using a self-similar transformation we solve the Riemann problem outside a neighborhood of the origin and then work inwards. Next a Goursant-type problem has to be solved to describe the interaction of waves up to the sonic circle. Inside it a system of composite elliptic-hyperbolic type is obtained, which may not always be solvable exactly. There an interesting local maximum principle can be shown. Finally, an exact partial solution is used for numerical comparisons.
Citation: Jiequan Li, Mária Lukáčová - MedviĎová, Gerald Warnecke. Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 559-576. doi: 10.3934/dcds.2003.9.559
 [1] Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153 [2] Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127 [3] Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361 [4] Jingwei Hu, Shi Jin. On kinetic flux vector splitting schemes for quantum Euler equations. Kinetic & Related Models, 2011, 4 (2) : 517-530. doi: 10.3934/krm.2011.4.517 [5] Cheng Wang, Jian-Guo Liu. Positivity property of second-order flux-splitting schemes for the compressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 201-228. doi: 10.3934/dcdsb.2003.3.201 [6] Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146 [7] Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871 [8] Amy Allwright, Abdon Atangana. Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 443-466. doi: 10.3934/dcdss.2020025 [9] Paolo Baiti, Helge Kristian Jenssen. Blowup in $\mathbf{L^{\infty}}$ for a class of genuinely nonlinear hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 837-853. doi: 10.3934/dcds.2001.7.837 [10] Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555 [11] Peng Zhang, Jiequan Li, Tong Zhang. On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 609-634. doi: 10.3934/dcds.1998.4.609 [12] Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419 [13] Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090 [14] Cheng Wang, Xiaoming Wang, Steven M. Wise. Unconditionally stable schemes for equations of thin film epitaxy. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 405-423. doi: 10.3934/dcds.2010.28.405 [15] Tatsien Li, Wancheng Sheng. The general multi-dimensional Riemann problem for hyperbolic systems with real constant coefficients. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 737-744. doi: 10.3934/dcds.2002.8.737 [16] Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 [17] Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735 [18] Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34 [19] Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic & Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019 [20] José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1

2018 Impact Factor: 1.143