1. Balsara, D., Shu, C.W.:Monotonicity preserving weighted essentially nonoscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405-452 (2000) 2. Ben-Artzi, M., Falcovitz, J.:A second order Godunov-type scheme for compressible fuid dynamics. J. Comput. Phys. 55, 1-32 (1984) 3. Castro, C.E., Toro, E.F.:Solvers for the high-order Riemann problem for hyperbolic balance laws. J. Comput. Phys. 227, 2481-2513 (2008) 4. Colella, P.:Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87, 171-200 (1990) 5. Dumbser, M.:Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations. Comput. Fluids 39, 60-76 (2010) 6. Dumbser, M., Balsara, D., Toro, E.F., Munz, C.D.:A unifed framework for the construction of one-step fnite-volume and discontinuous Galerkin schemes. J. Comput. Phys. 227, 8209-8253 (2008) 7. Dumbser, M., Enaux, C., Toro, E.F.:Finite volume schemes of very high order of accuracy for stif hyperbolic balance laws. J. Comput. Phys. 227, 3971-4001 (2008) 8. Glimm, J., Marshall, G., Plohr, B.:A generalized Riemann problem for quasi-one-dimensional gas fows. Adv. Appl. Math. 5, 1-30 (1984) 9. Goetz, C.R., Dumbser, M.:A novel solver for the generalized Riemann problem based on a simplifed LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation. J. Sci. Comput. 69, 805-840 (2016) 10. Goetz, C.R., Iske, A.:Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws. Math. Comput. 85, 35-62 (2016) 11. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.:Uniformly high order accuracy essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231-303 (1987) 12. Jiang, G.S., Shu, C.W.:Efcient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228 (1996) 13. Kall, J.:ADER Schemes for Systems of Conservation Laws on Networks. Technical report, Technische Universitat Kaiserslautern, (2015) 14. Kreiss, H.-O.:Über die stabilitätsdefnition für diferenzengleichungen die partielle diferentialgleichungen approximieren. BIT Numer. Math. 2, 153-181 (1962) 15. Krivodonova, L., Qin, R.:An analysis of the spectrum of the discontinuous Galerkin method. Appl. Numer. Math. 64, 1-18 (2013) 16. Le Floch, P., Raviart, P.A.:An asymptotic expansion for the solution of the generalized Riemann problem. Part 1:general theory. Ann. Inst. Henri Poincaré. Analyse non Lineáre 5(2), 179-207 (1988) 17. Le Floch, P., Tatsien, L.:A global asymptotic expansion for the solution of the generalized Riemann problem. Ann. Inst. Henri Poincaré. Analyse non Lineáre 3, 321-340 (1991) 18. Liu, X.D., Osher, S., Chan, T.:Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200-212 (1994) 19. Mengi, E., Overton, M.L.:Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix. IMA J. Numer. Anal. 25, 648-669 (2005) 20. Men'shov, I.S.:Increasing the order of approximation of Godunov's scheme using the generalized Riemann problem. USSR Comput. Math. Phys. 30(5), 54-65 (1990) 21. Montecinos, G.I.:Analytic Solutions for the Burgers Equation with Source Terms. arXiv e-prints, (2015) 22. Reddy, S.C., Trefethen, L.N.:Stability of the method of lines. Numerische Mathematik 62, 235-267 (1992) 23. Spijker, M.N., Tracogna, S., Welfert, B.D.:About the sharpeness of the stability estimates in the Kreiss matrix theorem. Math. Comput. 72, 697-713 (2002) 24. Suresh, A., Huynh, T.:Accurate monotonicity preserving scheme using Runge-Kutta time stepping. J. Comput. Phys. 136, 83-99 (1997) 25. Tan, S., Shu, C.W.:Inverse Lax-Wendrof procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229, 8144-8166 (2010) 26. Tan, S., Wang, C., Shu, C.W., Ning, J.:Efcient implementation of high order inverse Lax-Wendrof boundary treatment for conservation laws. J. Comput. Phys. 231, 2510-2527 (2012) 27. Tatsien, L., Wenci, Y.:Boundary-Value Problems for Quasi-linear Hyperbolic Systems. Duke University Mathematics Series (1985) 28. Titarev, V.A.:Derivative Riemann Problem and ADER Schemes. PhD thesis, Department of Mathematics, University of Trento, Italy (2005) 29. Titarev, V.A., Toro, E.F.:ADER:arbitrary high order Godunov approach. J. Sci. Comput. 17, 609-618 (2002) 30. Titarev, V.A., Toro, E.F.:Analysis of ADER and ADER-WAF schemes. IMA J. Numer. Anal. 27, 616-630 (2007) 31. Toro, E.F.:Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, New York (1997) 32. Toro, E.F.:Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, New York (2009) 33. Toro, E.F., Titarev, V.A.:Derivative Riemann solvers for systems of conservation laws and ADER methods. J. Comput. Phys. 212(1), 150-165 (2006) 34. Toro, E.F., Billett, S.J.:Centred TVD Schemes for Hyperbolic Conservation Laws. Technical Report MMU-9603, Department of Mathematics and Physics, Manchester Metropolitan University, UK, (1996) 35. Toro, E.F., Millington, R.C., Nejad, L.A.M.:Towards very high-order Godunov schemes. In:Toro, E.F. (ed.) Godunov Methods:Theory and Applications. Edited Review, pp. 905-937. Kluwer Academic, Dordrecht (2001) 36. Toro, E.F., Montecinos, G.I.:Implicit, semi-analytical solution of the generalised Riemann problem for stif hyperbolic balance laws. J. Comput. Phys. 303, 146-172 (2015) 37. Toro, E.F., Titarev, V.A.:Solution of the generalised Riemann problem for advection-reaction equations. Proc. R. Soc. Lond. A 458, 271-281 (2002) 38. Toro, E.F., Titarev, V.A.:TVD fuxes for the high-order ADER schemes. J. Sci. Comput. 24, 285-309 (2005) 39. Trefethen, L.N.:Finite Diference and Spectral Methods for Ordinary and Partial Diferential Equations. Technical report, Department of Computer Science and Center for Applied Mathematics, Cornell University, USA, (1996) 40. Trefethen, L.N., Embree, M.:Spectra and Pseudospectra:the Behaviour of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005) 41. Van Leer, B.:On the relation between the upwind-diferencing schemes of Godunov, Enguist-Osher and Roe. SIAM J. Sci. Stat. Comput. 5(1), 1-20 (1985) |