# zbMATH — the first resource for mathematics

A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics. (English) Zbl 1349.76551
Summary: The paper proposes and implements a third-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for one- and two-dimensional (1D & 2D) Euler equations in gas dynamics. It is an extension of the second-order accurate GRP scheme proposed in [M. Ben-Artzi et al., ibid. 218, No. 1, 19–43 (2006; Zbl 1158.76375)]. The approximate states in numerical fluxes of the third-order accurate GRP scheme are derived by using the higher-order WENO reconstruction of the initial data, the limiting values of the time derivatives of the solutions at the singularity point, and the Jacobian matrix. Besides the limiting values of the first-order time derivatives of fluid variables, the second-order time derivatives are also needed in developing the present GRP scheme and obtained by directly and analytically resolving the local GRP in the Eulerian formulation via two main ingredients, i. e. the Riemann invariants and Rankine-Hugoniot jump conditions. Unfortunately, for the sonic case that the transonic rarefaction wave appears in the GRP, the Jacobian matrix is singular on the sonic line. To this end, those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation. The 2D GRP scheme is implemented by using the third-order accurate time-splitting method. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed GRP scheme, in comparison to the second-order accurate GRP scheme.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76N15 Gas dynamics (general theory) 82C40 Kinetic theory of gases in time-dependent statistical mechanics 82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
Full Text:
##### References:
 [1] Ben-Artzi, M., The generalized Riemann problem for reactive flows, J. Comput. Phys., 81, 70-101, (1989) · Zbl 0668.76080 [2] Ben-Artzi, M.; Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys., 55, 1-32, (1984) · Zbl 0535.76070 [3] Ben-Artzi, M.; Falcovitz, J., Generalized Riemann problems in computational fluid dynamics, (2003), Cambridge University Press · Zbl 1017.76001 [4] Ben-Artzi, M.; Li, J. Q., Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem, Numer. Math., 106, 3, 369-425, (2007) · Zbl 1123.65082 [5] Ben-Artzi, M.; Li, J. Q.; Warnecke, G., A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys., 218, 19-43, (2006) · Zbl 1158.76375 [6] Birman, A.; Har’el, N. Y.; Falcovitz, J.; Ben-Artzi, M.; Feldman, U., Operator-split computation of 3-D symmetric flow, Comput. Fluid Dyn. J., 10, 37-43, (2001) [7] Borges, R.; Carmona, M.; Costa, B.; Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227, 3191-3211, (2008) · Zbl 1136.65076 [8] Falcovitz, J.; Alfandary, G.; Hanoch, G., A 2D conservation laws scheme for compressible flows with moving boundaries, J. Comput. Phys., 138, 83-102, (1997) · Zbl 0901.76044 [9] Godunov, S. K., A finite difference method for the numerical computation and discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47, 271-295, (1959) [10] Han, E.; Li, J. Q.; Tang, H. Z., An adaptive GRP scheme for compressible fluid flows, J. Comput. Phys., 229, 1448-1466, (2010) · Zbl 1329.76205 [11] Han, E.; Li, J. Q.; Tang, H. Z., Accuracy of the adaptive GRP scheme and the simulation of 2-D Riemann problems for compressible Euler equations, Commun. Comput. Phys., 10, 3, 577-606, (2011) · Zbl 1373.76130 [12] Kurganov, A.; Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differ. Equ., 18, 5, 584-608, (2002) · Zbl 1058.76046 [13] Lax, P. D.; Liu, X. D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19, 2, 319-340, (1998) · Zbl 0952.76060 [14] Li, J. Q.; Chen, G. X., The generalized Riemann problem method for the shallow water equations with bottom topography, Int. J. Numer. Methods Eng., 65, 834-862, (2006) · Zbl 1178.76249 [15] Liska, R.; Wendroff, B., Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM J. Sci. Comput., 25, 3, 995-1017, (2013) · Zbl 1096.65089 [16] Liu, N.; Tang, H. Z., A high-order accurate gas-kinetic scheme for one- and two-dimensional flow simulation, Commun. Comput. Phys., 15, 911-943, (2014) · Zbl 1373.76259 [17] Qian, J. Z.; Li, J. Q.; Wang, S. H., The generalized Riemann problems for compressible fluid flows: towards high order, J. Comput. Phys., 259, 358-389, (2014) · Zbl 1349.76379 [18] Schulz-Rinne, C. W.; Collins, J. P.; Glaz, H. M., Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput., 14, 6, 1394-1414, (1993) · Zbl 0785.76050 [19] Shi, J.; Zhang, Y. T.; Shu, C. W., Resolution of high order WENO schemes for complicated flow structures, J. Comput. Phys., 186, 690-696, (2003) · Zbl 1047.76081 [20] Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes. II, J. Comput. Phys., 83, 1, 32-78, (1989) · Zbl 0674.65061 [21] Sornborger, A. T.; Stewart, E. D., Higher-order methods for simulations on quantum computers, Phys. Rev. A, 60, 1956-1965, (1999) [22] Tang, H. Z.; Tang, T., Adaptive mesh methods for one- and two- dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41, 487-515, (2003) · Zbl 1052.65079 [23] Thalhammer, M.; Caliari, M.; Neuhauser, C., High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228, 822-832, (2009) · Zbl 1158.65340 [24] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (2009), Springer · Zbl 1227.76006 [25] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 1, 115-173, (1984) · Zbl 0573.76057 [26] Yang, Z. C.; He, P.; Tang, H. Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: one-dimensional case, J. Comput. Phys., 230, 22, 7964-7987, (2011) · Zbl 1408.76597 [27] Yang, Z. C.; Tang, H. Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: two-dimensional case, J. Comput. Phys., 231, 2116-2139, (2012) · Zbl 1408.76598 [28] Zhang, T.; Zheng, Y., Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal., 21, 3, 593-630, (1990) · Zbl 0726.35081 [29] Zhao, J.; Tang, H. Z., Runge-Kutta discontinuous Galerkin methods with WENO limiters for the special relativistic hydrodynamics, J. Comput. Phys., 242, 138-168, (2013) · Zbl 1314.76035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.