zbMATH — the first resource for mathematics

A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems. (English) Zbl 1307.76056
Summary: We present a fully second order implicit/explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems. The idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations. In this scope, the hydrodynamics equations are discretized explicitly making use of the capability of well-understood explicit schemes. On the other hand, the nonlinear heat conduction is solved implicitly. Such methods are often referred to as IMEX methods. The Jacobian-Free Newton Krylov (JFNK) method is applied to the problem in such a way as to render a nonlinearly iterated IMEX method. We solve three test problems in order to validate the numerical order of the scheme. For each test, we established second order time convergence. We support these numerical results with a modified equation analysis (MEA). The set of equations studied here constitute a base model for radiation hydrodynamics.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N15 Gas dynamics, general
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
Full Text: DOI
[1] Ascher, U.M.; Ruuth, S.J.; Spiteri, R.J., Implicit – explicit runge – kutta methods for time dependent partial differential equations, Appl. numer. math., 25, 151-167, (1997) · Zbl 0896.65061
[2] Ascher, U.M.; Ruuth, S.J.; Wetton, B., Implicit-explicit methods for time dependent PDE’s, SIAM J. numer. anal., 32, 797-823, (1995) · Zbl 0841.65081
[3] Bates, J.W.; Knoll, D.A.; Rider, W.J.; Lowrie, R.B.; Mousseau, V.A., On consistent time-integration methods for radiation hydrodynamics in the equilibrium diffusion limit: low-energy-density regime, J. comput. phys., 167, 99-130, (2001) · Zbl 1052.76041
[4] Bowers, R.L.; Wilson, J.R., Numerical modeling in applied physics and astrophysics, (1991), Jones and Bartlett Boston · Zbl 0786.76001
[5] Dai, W.; Woodward, P.R., Numerical simulations for radiation hydrodynamics. I. diffusion limit, J. comput. phys., 142, 182, (1998) · Zbl 0933.76057
[6] Gottlieb, S.; Shu, C.W., Total variation diminishing runge – kutta schemes, Math. comput., 221, 73-85, (1998) · Zbl 0897.65058
[7] Gottlieb, S.; Shu, C.W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM rev., 43-1, 89-112, (2001) · Zbl 0967.65098
[8] Kadioglu, S.Y.; Knoll, D.A.; Oliveria, C., Multi-physics analysis of spherical fast burst reactors, Nucl. sci. eng., 163, 1-12, (2009)
[9] Kelley, C.T., Solving nonlinear equations with newton’s method, SIAM, (2003) · Zbl 1031.65069
[10] Knoll, D.A.; Keyes, D.E., Jacobian-free Newton Krylov methods: a survey of approaches and applications, J. comput. phys., 193, 357-397, (2004) · Zbl 1036.65045
[11] Leveque, R.J., Finite volume methods for hyperbolic problems, (1998), Cambridge University Press, (Texts in Applied Mathematics)
[12] Lowrie, R.B.; Morel, J.E.; Hittinger, J.A., The coupling of radiation and hydrodynamics, Astrophys. J., 521, 432, (1999)
[13] Marshak, R.E., Effect of radiation on shock wave behavior, Phys. fluids, 1, 24-29, (1958) · Zbl 0081.41601
[14] Reid, J.K., On the methods of conjugate gradients for the solution of large sparse systems of linear equations. large sparse sets of linear equations, (1971), Academic Press New York · Zbl 0229.65032
[15] Rider, W.J.; Knoll, D.A., Time step size selection for radiation diffusion calculations, J. comput. phys., 152-2, 790-795, (1999) · Zbl 0940.65101
[16] Saad, Y., Iterative methods for sparse linear systems, SIAM, (2003) · Zbl 1002.65042
[17] Shestakov, A.I., Time-dependent simulations of point explosions with heat conduction, Phys. fluids, 11, 1091-1095, (1999) · Zbl 1147.76497
[18] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[19] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061
[20] Strikwerda, J.C., Finite difference schemes partial differential equations, (1989), Wadsworth & Brooks/Cole, Advance Books & Software Pacific Grove, CA · Zbl 0681.65064
[21] Thomas, J.W., Numerical partial differential equations I (finite difference methods), (1998), Springer-Verlag New York, (Texts in Applied Mathematics)
[22] Thomas, J.W., Numerical partial differential equations II (conservation laws and elliptic equations), (1999), Springer-Verlag New York, (Texts in Applied Mathematics) · Zbl 0927.65109
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.