zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A modified higher-order Godunov’s scheme for stiff source conservative hydrodynamics. (English) Zbl 1117.76039
Summary: We present an efficient second-order accurate scheme to treat stiff source terms within the framework of higher-order Godunov methods. We employ Duhamel formula to devise a modified predictor step which accounts for the effects of stiff source terms on the conservative fluxes and recovers the correct isothermal behavior in the limit of an infinite cooling/reaction rate. Source term effects on the conservative quantities are fully accounted for by means of a one-step, second-order accurate semi-implicit corrector scheme based on the deferred correction method of {\it A. Dutt} et al. [BIT 40, No. 2, 241--266 (2000; Zbl 0959.65084)]. We demonstrate the accurate, stable and convergent results of the proposed method through a set of benchmark problems for a variety of stiffness conditions and source types.

MSC:
76M12Finite volume methods (fluid mechanics)
76N15Gas dynamics, general
WorldCat.org
Full Text: DOI
References:
[1] Berger, M. J.; Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. comput. Phys. 82, 64-84 (1989) · Zbl 0665.76070
[2] Caflisch, R. E.; Jin, S.; Russo, G.: Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Sci. comput. 34, No. 1, 246-281 (1997) · Zbl 0868.35070
[3] Chen, G. Q.; Levermore, C.; Liu, T.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. pure appl. Math. 47, 787-830 (1994) · Zbl 0806.35112
[4] Colella, P.: Multidimensional upwind methods for hyperbolic conservation laws. J. comput. Phys. 82, 64-84 (1989) · Zbl 0665.76070
[5] Dutt, A.; Greengard, L.; Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. Bit 40, 241-266 (2000) · Zbl 0959.65084
[6] Field, G. B.: Thermal instability. Astrophys. J. 142, 531 (1965)
[7] Jin, S.: Runge -- Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. comput. Phys. 122, 51-67 (1995) · Zbl 0840.65098
[8] Jin, S.; Levermore, D.: Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. comput. Phys. 126, 449-467 (1996) · Zbl 0860.65089
[9] Jin, S.; Pareschi, L.; Toscani, G.: Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations. SIAM J. Sci. comput. 35, No. 6, 2406-2439 (1998) · Zbl 0938.35097
[10] F. Miniati, P. Colella, Block structured adaptive mesh and time refinement for hybrid, hyperbolic+n-body systems, J. Comput. Phys. [e-print: astro-ph/0608156]. · Zbl 1128.85007
[11] Miniati, F.; Ryu, D.; Kang, H.; Jones, T. W.; Cen, R.; Ostriker, J.: Properties of cosmic shock waves in large-scale structure formation. Astrophys. J. 542, 608-621 (2000)
[12] Minion, M.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Comm. math. Sci. 1, 471-500 (2003) · Zbl 1088.65556
[13] Pareschi, L.; Russo, G.: Implicit --- explicit Runge -- Kutta schemes and applications to hyperbolic systems with relaxation. SIAM J. Sci. comput. 25, No. 1, 129-155 (2005) · Zbl 1203.65111
[14] Pember, R. B.: Numerical methods for hyperbolic conservation laws with stiff relaxation II: Higher-order Godunov methods. SIAM J. Sci. comput. 14, No. 4, 824-859 (1993) · Zbl 0812.65083
[15] Roe, P. L.; Hittinger, A. F.: Toward Godunov-type methods for hyperbolic conservation laws with stiff relaxation. Godunov methods: theory and applications, 725-744 (2001) · Zbl 1036.76041
[16] Saltzman, J.: An unsplit 3D upwind method for hyperbolic conservation laws. J. comput. Phys. 115, 153-168 (1994) · Zbl 0813.65111
[17] Trebotich, D.; Colella, P.; Miller, G. H.: A stable and convergent scheme for viscoelastic flow in contraction channels. J. comput. Phys. 205, No. May, 315-342 (2005) · Zbl 1087.76005
[18] Vincenti, W. G.; Kruger, C. H.: Introduction to physical gas dynamics. (1965)
[19] Withman, G. B.: Linear and non-linear waves. (1974)
[20] Woodward, P. R.; Colella, P.: Numerical simulations of two-dimensional fluid flow with strong shocks. J. comput. Phys. 54, 115-173 (1984) · Zbl 0573.76057