# zbMATH — the first resource for mathematics

Mixed analytical/numerical method for flow equations with a source term. (English) Zbl 1048.76044
Summary: A mixed analytical/numerical approach is studied for flow problems described by partial differential equations with source terms which are analytically integrable and which may involve a time scale ($$S$$-scale) much smaller than the mean flow time scale ($$M$$-scale). A rigorous error analysis based on the modified equation is conducted for a linear model equation, and it is shown, both analytically and numerically, that the mixed scheme is more accurate than a conventional numerical method. Most interestingly, the mixed approach has a good accuracy for the $$M$$-scale structure even though the time step is larger than the $$S$$-scale, while a conventional scheme fails to work in this case by producing errors of order $$O(1)$$ or larger.

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76N15 Gas dynamics, general
##### Keywords:
hyperbolic system; error analysis
HE-E1GODF
Full Text:
##### References:
 [1] Bao, W.Z.; Jin, S., The random projection method for hyperbolic conservation laws with stiff reaction terms, J. computat. phys., 163, 216-248, (2000) · Zbl 0966.65073 [2] Bermudez, L.; Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput fluids, 23, 1049-1071, (1994) · Zbl 0816.76052 [3] Boris, J.P.; Oran, E.S., Numerical simulation of reactive flow, (1987), Elsevier Amsterdam · Zbl 0762.76098 [4] E, W.N., Homogenization of scalar conservation laws with oscillatory forcing terms, SIAM J appl math, 52, 959-972, (1992) · Zbl 0755.35068 [5] Caflisch, R.E.; Jin, S.; Russo, G., Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J numer anal, 34, 246-281, (1997) · Zbl 0868.35070 [6] Chalabi, A., Stable upwind schemes for conservation laws with source term, IMA J numer anal, 12, 217-241, (1992) · Zbl 0754.65077 [7] Chalabi, A., An error bound for the polygonal approximation of hyperbolic conservation laws with source terms, Comput math appl, 32, 59-63, (1996) · Zbl 0865.35083 [8] Chalabi, A., On the convergence of numerical schemes for hyperbolic conservation laws with stiff source terms, Math computat, 66, 527-545, (1997) · Zbl 0865.35084 [9] Glimm, J.; Marshall, G.; Plohr, B., A generalized Riemann problem for quasi-one-dimensional gas flows, Adv appl math, 5, 1-30, (1984) · Zbl 0566.76056 [10] Griffiths, D.F.; Stuart, A.M.; Yee, H.C., Numerical wave propagation in an advection equation with a nonlinear source term, SIAM J numer anal, 29, 1244-1260, (1992) · Zbl 0759.65060 [11] Gosse, L., A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math models meth appl sci, 11, 339-365, (2001) · Zbl 1018.65108 [12] Helzel, C.; Leveque, R.J.; Warnecke, G., A modified fractional step method for the accurate approximation of detonation waves, SIAM J scientific comput, 22, 1489-1510, (2000) · Zbl 0983.65105 [13] Hui WH, Koudriakov S. Computation of the shallow water equations using the unified coordinates, accepted · Zbl 1067.76011 [14] Jin, S., Runge – kutta methods for hyperbolic conservation laws with stiff relaxation terms, J computat phys, 123, 51-67, (1995) · Zbl 0840.65098 [15] Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms, Math model numer anal, 35, 631-646, (2001) · Zbl 1001.35083 [16] Jin, S.; Xin, Z.P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun pure appl math, XLVIII, 235-276, (1995) · Zbl 0826.65078 [17] Launder, B.E.; Spalding, D.B., Mathematical models of turbulence, (1972), Academic Press New York · Zbl 0288.76027 [18] van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J computat phys, 32, 101-136, (1979) · Zbl 1364.65223 [19] Lerat, A.; Sides, J., Numerical simulation of unsteady transonic flows using the Euler equations in integral form, Israel J technol, 17, 302, (1979) · Zbl 0458.76047 [20] LeVeque, R.J.; Yee, H.C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J computat phys, 86, 187-210, (1990) · Zbl 0682.76053 [21] Leveque, R.J., Balancing source terms and flux gradients in high resolution Godunov methods, J computat phys, 146, 346-365, (1998) · Zbl 0931.76059 [22] Liu, F.; Zheng, X.Q., A strongly coupled time-marching method for solving the navier – stokes and k-omega turbulence model equations with multigrid, J computat phys, 128, 289-300, (1996) · Zbl 0862.76064 [23] Merci, B.; Steelant, J.; Vierendeels, J.; Riemslagj, K.; Dick, E., Computational treatment of source terms in two-equation turbulence models, Aiaa j, 38, 2085-2093, (2000) [24] Pember, R.B., Numerical methods for hyperbolic conservation laws with stiff relaxation I: spurious solutions, SIAM J appl math, 53, 1293-1330, (1993) · Zbl 0787.65062 [25] Pember, R.B., Numerical methods for hyperbolic conservation laws with stiff relaxation I: high order Godunov methods, SIAM J scientific comput, 14, 824-859, (1993) · Zbl 0812.65083 [26] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J computat phys, 43, 357-372, (1981) · Zbl 0474.65066 [27] Roe, P.L., Upwinding differencing schemes for hyperbolic conservation laws with source terms, Proceedings of the 1st international conference for hyperbolic problems, 2, 41-56, (1986) [28] Sod, G.A., A numerical study of a converging shock, J fluid mech, 83, 785-794, (1997) · Zbl 0366.76055 [29] Strang, G., On the construction and comparison of difference schemes, SIAM J numer anal, 5, 506-517, (1968) · Zbl 0184.38503 [30] Tang, T.; Wang, J.-H., Convergence of MUSCL relaxing schemes to the relaxed schemes for conservation laws with stiff source terms, J scientific comput, 15, 173-196, (2000) · Zbl 0982.65101 [31] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, (1999), Springer Berlin · Zbl 0923.76004 [32] Vazquez-Cendon, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J computat phys, 148, 497-526, (1998 1999) [33] Xu, K., A gas-kinetic scheme for the Euler equations with heat transfer, SIAM J scientific comput, 20, 1317-1335, (1999) · Zbl 0959.76064 [34] Xu K. A well-balanced gas-kinetic scheme for the shallow water equations with source term, preprint [35] Yanenko, N.N., The method of fractional steps, (1971), Springer-Verlag New York · Zbl 0209.47103 [36] Yee HC, Sweby PK. Nonlinear dynamics and numerical uncertainties in CFD, Tech. Report TM 110398, NASA Ames Research Center, Moffett Field, USA, 1996 [37] Yee, H.C.; Shnn, J., Semi-implicit and fully implicit shock capturing methods for non-equilibrium flows, Aiaa j, 27, 299-307, (1989)
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.