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An improved Roe solver for high order reconstruction schemes. (English) Zbl 07211867
Summary: Roe solver is one of the most popular upwind schemes in computational fluid dynamics which is first-order accurate in space. However, the amount of numerical dissipation gets larger than necessary for high order numerical schemes and then becomes the dominant term of the numerical error. This paper proposes a modified Roe solver that can use high order schemes with high stability based on splitting of the upwind term. The novel algorithm is proposed to compute the dissipation term of Roe flux function using low and high reconstruction schemes for computing acoustic and entropy waves information, respectively. It changes the original Roe solver dissipation term to improve the level of accuracy. The third-order WENO-Z and MUSCL schemes are adopted as high reconstruction schemes. Finite difference approach is used to simulate Euler equations with the original and modified Roe solvers. The results of four shock tube problems of Sod, Lax, Mach 3, and supersonic test cases are compared with the exact solution of the Riemann problem. It is found that the proposed Roe solver increases the accuracy and robustness of the original Roe solver, especially around the expansion waves, contact discontinuities, and shock waves.
MSC:
76 Fluid mechanics
Software:
AUSM
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[1] Einfeldt, B.; Munz, C.-D.; Roe, P. L.; Sjögreen, B., On godunov-type methods near low densities, J Comput Phys, 92, 2, 273-295 (1991) · Zbl 0709.76102
[2] Osher, S.; Chakravarthy, S., Upwind schemes and boundary conditions with applications to euler equations in general geometries, J Comput Phys, 50, 3, 447-481 (1983) · Zbl 0518.76060
[3] Roe, P. L., Approximate riemann solvers, parameter vectors, and difference schemes, J Comput Phys, 43, 2, 357-372 (1981) · Zbl 0474.65066
[4] Jameson, A.; Shmidt, W.; Turkel, E., Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes (2012)
[5] SWANSON, R.; TURKEL, E., Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations (2014)
[6] Steger, J. L.; Warming, R., Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods, J Comput Phys, 40, 2, 263-293 (1981) · Zbl 0468.76066
[7] Van Leer, B., Flux-vector splitting for the Euler equation, 80-89 (1997), Springer
[8] Liou, M.-S., A sequel to ausm: ausm+, J Comput Phys, 129, 2, 364-382 (1996) · Zbl 0870.76049
[9] Kim, K. H.; Lee, J. H.; Rho, O. H., An improvement of ausm schemes by introducing the pressure-based weight functions, Computers & fluids, 27, 3, 311-346 (1998) · Zbl 0964.76064
[10] Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89, 3, 271-306 (1959) · Zbl 0171.46204
[11] Roe, P. L., Approximate riemann solvers, parameter vectors, and difference schemes, J Comput Phys, 135, 2, 250-258 (1997) · Zbl 0890.65094
[12] Weiss, J. M.; Smith, W. A., Preconditioning applied to variable and constant density flows, AIAA journal, 33, 11, 2050-2057 (1995) · Zbl 0849.76072
[13] Turkel, E., Preconditioning techniques in computational fluid dynamics, Annu Rev Fluid Mech, 31, 1, 385-416 (1999)
[14] Li, X.-s.; Gu, C.-w., An all-speed roe-type scheme and its asymptotic analysis of low mach number behaviour, J Comput Phys, 227, 10, 5144-5159 (2008) · Zbl 1388.76207
[15] Huang, D., Preconditioned dual-time procedures and its application to simulating the flow with cavitations, J Comput Phys, 223, 2, 685-689 (2007) · Zbl 1111.76037
[16] Park, S. H.; Lee, J. E.; Kwon, J. H., Preconditioned hlle method for flows at all mach numbers, AIAA journal, 44, 11, 2645-2653 (2006)
[17] Li, X.-s.; Gu, C.-w.; Xu, J.-z., Development of roe-type scheme for all-speed flows based on preconditioning method, Computers & Fluids, 38, 4, 810-817 (2009) · Zbl 1242.76171
[18] Thornber, B.; Drikakis, D., Numerical dissipation of upwind schemes in low mach flow, Int J Numer Methods Fluids, 56, 8, 1535-1541 (2008) · Zbl 1136.76034
[19] Kim, S.-s.; Kim, C.; Rho, O.-H.; Hong, S. K., Cures for the shock instability: development of a shock-stable roe scheme, J Comput Phys, 185, 2, 342-374 (2003) · Zbl 1062.76538
[20] Qu, F.; Yan, C.; Sun, D.; Jiang, Z., A new roe-type scheme for all speeds, Computers & Fluids, 121, 11-25 (2015) · Zbl 1390.76686
[21] Kermani, M.; Plett, E., Modified entropy correction formula for the roe scheme, Proceedings of the 39th aerospace sciences meeting and exhibit, 83 (2012)
[22] Quirk, J. J., A contribution to the great Riemann solver debate, 550-569 (1997), Springer
[23] Ren, Y.-X., A robust shock-capturing scheme based on rotated riemann solvers, Computers & Fluids, 32, 10, 1379-1403 (2003) · Zbl 1034.76035
[24] Nishikawa, H.; Kitamura, K., Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid riemann solvers, J Comput Phys, 227, 4, 2560-2581 (2008) · Zbl 1388.76185
[25] Lin, H.-C., Dissipation additions to flux-difference splitting, J Comput Phys, 117, 1, 20-27 (1995) · Zbl 0836.76061
[26] Liou, M.-S., Mass flux schemes and connection to shock instability, J Comput Phys, 160, 2, 623-648 (2000) · Zbl 0967.76062
[27] Van Leer, B., Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov’s method, J Comput Phys, 32, 1, 101-136 (1979) · Zbl 1364.65223
[28] Van Albada, G.; Van Leer, B.; Roberts, W., A comparative study of computational methods in cosmic gas dynamics, 95-103 (1997), Springer
[29] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J Comput Phys, 115, 1, 200-212 (1994) · Zbl 0811.65076
[30] Borges, R.; Carmona, M.; Costa, B.; Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J Comput Phys, 227, 6, 3191-3211 (2008) · Zbl 1136.65076
[31] Don, W.-S.; Borges, R., Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes, J Comput Phys, 250, 347-372 (2013) · Zbl 1349.65285
[32] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted eno schemes, J Comput Phys, 126, 1, 202-228 (1996) · Zbl 0877.65065
[33] Euler, L., Institutionum calculi integralis, 1 (1824), impensis Academiae imperialis scientiarum
[34] Gottlieb, S.; Shu, C.-W., Total variation diminishing runge-kutta schemes, Mathematics of computation of the American Mathematical Society, 67, 221, 73-85 (1998) · Zbl 0897.65058
[35] Harten, A., High resolution schemes for hyperbolic conservation laws, J Comput Phys, 49, 3, 357-393 (1983) · Zbl 0565.65050
[36] Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J Comput Phys, 27, 1, 1-31 (1978) · Zbl 0387.76063
[37] Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun Pure Appl Math, 7, 1, 159-193 (1954) · Zbl 0055.19404
[38] Arora, M.; Roe, P. L., A well-behaved tvd limiter for high-resolution calculations of unsteady flow, J Comput Phys, 132, 1, 3-11 (1997) · Zbl 0878.76045
[39] Wesseling, P., Principles of computational fluid dynamics, 29 (2009), Springer Science & Business Media · Zbl 1185.76005
[40] Glaister, P., An approximate linearised riemann solver for the three-dimensional euler equations for real gases using operator splitting, J Comput Phys, 77, 2, 361-383 (1988) · Zbl 0644.76088
[41] https://www.reading.ac.uk/web/files/maths/NA_Report_12-86.pdf.
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