×

zbMATH — the first resource for mathematics

High-order well-balanced schemes and applications to non-equilibrium flow. (English) Zbl 1261.76024
Summary: The appearance of the source terms in modeling non-equilibrium flow problems containing finite-rate chemistry or combustion poses additional numerical difficulties beyond that for solving non-reacting flows. A well-balanced scheme, which can preserve certain non-trivial steady state solutions exactly, may help minimize some of these difficulties. In this paper, a simple one-dimensional non-equilibrium model with one temperature is considered. We first describe a general strategy to design high-order well-balanced finite-difference schemes and then study the well-balanced properties of the high-order finite-difference weighted essentially non-oscillatory (WENO) scheme, modified balanced WENO schemes and various total variation diminishing (TVD) schemes. The advantages of using a well-balanced scheme in preserving steady states and in resolving small perturbations of such states will be shown. Numerical examples containing both smooth and discontinuous solutions are included to verify the improved accuracy, in addition to the well-balanced behavior.

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
76V05 Reaction effects in flows
Software:
HLLE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, J.D., Computational fluids dynamics, (1995), McGraw-Hill New York
[2] B. Engquist, B. Sjögreen, Robust Difference Approximation for Stiff Inviscid Detonation Waves, Report CAM 91-05, Dept. of Math., UCLA, 1991.
[3] Gascón, Ll.; Corberán, J.M., Construction of second-order TVD schemes for nonhomogeneous hyperbolic conservation laws, J. comp. phys., 172, 261-297, (2001) · Zbl 0991.65072
[4] P.A. Gnoffo, R.N. Gupta, J.L. Shinn, Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium, NASA Technical Paper 2867, 1989, 1-58.
[5] Greenberg, J.M.; LeRoux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. numer. anal., 33, 1-16, (1996) · Zbl 0876.65064
[6] Griffiths, D.F.; Stuart, A.M.; Yee, H.C., Numerical wave propagation in hyperbolic problems with nonlinear source terms, SIAM J. numer. anal., 29, 1244-1260, (1992) · Zbl 0759.65060
[7] Grossman, B.; Cinnella, P., Flux-split algorithms for flows with non-equilibrium chemistry and vibrational relaxation, J. comp. phys., 88, 131-168, (1990) · Zbl 0703.76047
[8] Harten, A.; Lax, P.D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35-61, (1983) · Zbl 0565.65051
[9] Hubbard, M.E.; Garcia-Navarro, P., Flux difference splitting and the balancing of source terms and flux gradients, J. comp. phys., 165, 89-125, (2000) · Zbl 0972.65056
[10] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comp. phys., 126, 202-228, (1996) · Zbl 0877.65065
[11] Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms, Math. modell. numer. anal. (M2AN), 35, 631-645, (2001) · Zbl 1001.35083
[12] Kennedy, C.A.; Carpenter, M.H., Additive runge – kutta schemes for convection – diffusion – reaction equations, Appl. numer. math., 44, 139-181, (2003) · Zbl 1013.65103
[13] Kurganov, A.; Levy, D., Central-upwind schemes for the saint – venant system, Math. modell. numer. anal. (M2AN), 36, 397-425, (2002) · Zbl 1137.65398
[14] Lafon, A.; Yee, H.C., Dynamical approach study of spurious steady-state numerical solutions for non-linear differential equations, part III: the effects of non-linear source terms in reaction – convection equations, Comp. fluid dyn., 6, 1-36, (1996)
[15] Lafon, A.; Yee, H.C., Dynamical approach study of spurious steady-state numerical solutions for non-linear differential equations, part IV: stability vs. numerical treatment of non-linear source terms, Comp. fluid dyn., 6, 89-123, (1996)
[16] LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. comp. phys., 146, 346-365, (1998) · Zbl 0931.76059
[17] LeVeque, R.J.; Yee, H.C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. comp. phys., 86, 187-210, (1990) · Zbl 0682.76053
[18] Noelle, S.; Xing, Y.; Shu, C.-W., High order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. comp. phys., 226, 29-58, (2007) · Zbl 1120.76046
[19] Rebollo, T.C.; Delgado, A.D.; Nieto, E.D.F., A family of stable numerical solvers for the shallow water equations with source terms, Comp. meth. appl. mech. eng., 192, 203-225, (2003) · Zbl 1083.76557
[20] Ricchiuto, M.; Abgrall, R.; Deconinck, H., Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes, J. comp. phys., 222, 287-331, (2007) · Zbl 1216.76051
[21] Ricchiuto, M.; Bollermann, A., Stabilized residual distribution for shallow water simulations, J. comp. phys., 228, 1071-1115, (2009) · Zbl 1330.76097
[22] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comp. phys., 43, 357-372, (1981) · Zbl 0474.65066
[23] G. Russo, Central schemes for balance laws, in: Proceedings of the VIII International Conference on Nonlinear Hyperbolic Problems, Magdeburg, 2000. · Zbl 0982.65093
[24] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (), 325-432 · Zbl 0927.65111
[25] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comp. phys., 77, 439-471, (1988) · Zbl 0653.65072
[26] Xing, Y.; Shu, C.-W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J. comp. phys., 208, 206-227, (2005) · Zbl 1114.76340
[27] Xing, Y.; Shu, C.-W., High order well-balanced finite difference WENO schemes for a class of hyperbolic systems with source terms, J. sci. comp., 27, 477-494, (2006) · Zbl 1115.76059
[28] Xu, K., A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J. comp. phys., 178, 533-562, (2002) · Zbl 1017.76071
[29] Yee, H.C., Construction of explicit and implicit symmetric TVD schemes and their applications, J. comp. phys., 68, 151-179, (1987) · Zbl 0621.76026
[30] H.C. Yee, A class of high-resolution explicit and implicit shock-capturing methods, VKI lecture series 1989-04, March 6-10, 1989; NASA TM-101088, Feb. 1989, 1989.
[31] Yee, H.C.; Harten, A., Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates, Aiaa j., 25, 266-274, (1987)
[32] Yee, H.C.; Shinn, J.L., Semi-implicit and fully implicit shock-capturing methods for nonequilibrium flows, Aiaa j., 225, 910-934, (1989)
[33] Yee, H.C.; Sjögreen, B., Development of low dissipative high order filter schemes for multiscale navier – stokes/MHD systems, J. comp. phys., 68, 151-179, (2007) · Zbl 1343.76053
[34] Yee, H.C.; Sweby, P.K.; Griffiths, D.F., Dynamical approach study of spurious steady-state numerical solutions for non-linear differential equations, part I: the dynamics of time discretizations and its implications for algorithm development in computational fluid dynamics, J. comp. phys., 97, 249-310, (1991) · Zbl 0760.65087
[35] Zhou, J.G.; Causon, D.M.; Mingham, C.G.; Ingram, D.M., The surface gradient method for the treatment of source terms in the shallow-water equations, J. comp. phys., 168, 1-25, (2001) · Zbl 1074.86500
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.