Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Abramovici et al. Graviational wave astrophysics. In E. W. Kolb and R. D. Peccei, editors, Particle and Nuclear Astrophysics and Cosmology in the Next Millennium, Proc. 1994 Snowmass Summer School, pages 398–425. World Scientific, 1994.
D. A. Anderson, J. C. Tannehill, and R. H. Pletcher. Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, 1984.
A. M. Anile. Relativistic Fluids and Magneto-Fluids. Cambridge University Press, Cambridge, 1989.
P. Anninos, G. Daues, J. Massó, E. Seidel, and W.-M. Suen. Horizon boundary conditions in numerical relativity. Physical Review D, 51:5562–5578, 1995.
P. C. Argyres. Universality and scaling in black hole formation. In E. W. Kolb and R. D. Peccei, editors, Particle and Nuclear Astrophysics and Cosmology in the Next Millennium, Proc. 1994 Snowmass Summer School, pages 447–449. World Scientific, 1994.
R. Arnowitt, S. Deser, and C. W. Misner. The dynamics of general relativity. In L. Witten, editor, Gravitation: An Introduction to Current Work, pages 227–265, New York, 1962. Wiley.
M. Arora and P. L. Roe. On postshock oscillations due to shock capturing schemes in unsteady flow. J. Comput. Phys., 130:1–24, 1997.
N. Aslan. Computational investigations of ideal MHD plasmas with discontinuities. PhD thesis, University of Michigan, Nuclear Eng. Dept., 1993.
N. Aslan. Numerical solutions of one-dimensional MHD equations by a fluctuation approach. Int. J. Numer. Meth. Fluids, 22:569–580, 1996.
N. Aslan. Two dimensional solutions of MHD equations with a modified Roe’s method. Int. J. Numer. Meth. Fluids, in print, 1996.
N. Aslan and T. Kammash. A Riemann solver for two dimensional MHD equations. Int. J. Numer. Meth. Fluids, to appear, 1997.
Martin Bailyn. A Survey of Thermodynamics. American Institute of Phyisics Press, Woodbury, NY, 1994.
D. Balsara. Modern schemes for solving hyperbolic conservation laws of interest in computational astrophysics on parallel machines. In D. A. Clarke, editor, Halifax Conference on Computational Astrophysics, 1996.
D. S. Balsara. Riemann solver for relativistic hydrodynamics. J. Comput. Phys., 114:284–297, 1994.
D. S. Balsara. Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics. Astrophys. J., to appear.
F. Banyuls, J. A. Font, J. M. Ibáñez, J. M. MartÃ, and J. A. Miralles. Numerical 3 + 1 general-relativistic hydrodynamics: a local characteristic approach. Astrophysical J., pages 221–231, 1997.
A. A. Barmin, A. G. Kulikovskiy, and N. V. Pogorelov. Shock-capturing approach and nonevolutionary solutions in magnetohydrodynamics. J. Comput. Phys., 126:77–90, 1996.
E. Battaner. Astrophysical Fluid Dynamics. Cambridge University Press, Cambridge, 1996.
J. Bell, M. J. Berger, J. Saltzman, and M. Welcome. Three dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Stat., 15:127–138, January 1994.
J. B. Bell, P. Colella, and H. M. Glaz. A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys., 85:257–283, 1989.
J. B. Bell, P. Colella, and J. R. Trangenstein. Higher order Godunov methods for general systems of hyperbolic conservation laws. J. Comput. Phys., 82:362, 1989.
J. B. Bell, C. N. Dawson, and G. R. Shubin. An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions. J. Comput. Phys., 74:1–24, 1988.
J. B. Bell and D. L. Marcus. A second-order projection method for variable-density flows. J. Comput. Phys., 101:334–348, 1992.
M. Ben-Artzi. The generalized Riemann problem for reactive flows. J. Comput. Phys., 81:70–101, 1989.
M. Berger. Adaptive mesh refinement for hyperbolic partial differential equations. PhD thesis, Computer Science Deptartment, Stanford University, 1982.
M. Berger and A. Jameson. Automatic adaptive grid refinement for the Euler equations. AIAA J., 23:561–568, 1985.
M. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53:484–512, 1984.
M. J. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82:64–84, 1989.
M. J. Berger and R. J. LeVeque. AMRCLAW software. A test version is available at http://www.amath.washington.edu/~rj1/amrclaw/.
M. J. Berger and R. J. LeVeque. Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal., to appear.
P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for collision processes in gases I. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94:511, 1954.
J. M. Blondin and B. S. Marks. Evolution of cold shock-bounded slabs. New Astronomy, 1:235–244, 1996.
C. Bona, J.M. Ibáñez, J.M. MartÃ, and J. Massó. Shock capturing methods in 1D numerical relativity. In F. Chinea, editor, Gravitation and General Relativity: rotating bodies and other topics, Lecture Notes in Physics, Vol. 423. Springer-Verlag, 1993.
C. Bona and J. Massó. A hyperbolic evolution system for numerical relativity. Phys. Rev. Letters, 68:1097, 1992.
C. Bona and J. Massó. Numerical relativity: Evolving spacetime. International Journal of Modern Physics C: Phys. and Computers, 4:883, 1993.
C. Bona, J. Massó, E. Seidel, and J. Stela. A new formalism for numerical relativity. Phys. Rev. Lett., 75:600, 1995.
J. P. Boris and D. L. Book. Flux corrected transport I, SHASTA, a fluid transport algorithm that works. J. Comput. Phys., 11:38–69, 1973.
A. Bourlioux. Numerical Study of Unstable Detonations. PhD thesis, Princeton University, June, 1991.
A. Bourlioux and A. J. Majda. Theoretical and numerical structure of unstable detonations. Phil. Trans. R. Soc. Lond. A, 350:29–68, 1995.
J. U. Brackbill and D. C. Barnes. The effect of nonzero ∇·B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys., 35:426–430, 1980.
W. L. Briggs. A Multigrid Tutorial. SIAM, 1987.
M. Brio and P. Rosenau. Stability of shock waves for 3 x 3 model MHD equations. Fourth Intl. Conf. Hyperbolic Prob., Taormina, 1992.
M. Brio and P. Rosenau. Evolution of the fast-intermediate shock wave in an MHD model problem. preprint, 1997.
M. Brio and C. C. Wu. An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys., 75:400–422, 1988.
Tung Chang and Ling Hsiao. The Riemann Problem and Interaction of Waves in Gas Dynamics. John Wiley and Sons Inc., New York, 1989.
M. Choptuik. Experiences with an adaptive mesh refinement algorithm in numerical relativity. In C. R. Evans, L. S. Finn, and D. W. Hobill, editors, Frontiers in Numerical Relativity. Cambridge University Press, 1989.
M. Choptuik. Consistency of finite-difference solutions of Einstein’s equations. Phys. Rev. D, 44:3124–3135, 1991.
M. Choptuik. Universality and scaling in gravitational collapse of a massless scalar field. Phys. Rev. Lett., 70:9–12, 1993.
M. W. Choptuik. Critical behaviour in massless scalar field collapse. In R. d’Inverno, editor, Approaches to Numerical Relativity, page 202. Cambridge University Press, 1992.
Y. Choquet-Bruhat and J. W. York, Jr. Geometrical well-posed systems for the Einstein equations. C. R. Acad. Sci. Paris, A321:1089–1095, 1995.
A. J. Chorin. Numerical solution of the Navier-Stokes equations. Math. Comp., 22:745–762, 1968.
A. J. Chorin. Random choice solution of hyperbolic systems. J. Comput. Phys., 22:517–533, 1976.
A. J. Chorin and J. E. Marsden. A Mathematical Introduction to Fluid Mechanics. Springer-Verlag, 1979.
P. Colella. Glimm’s method for gas dynamics. SIAM J. Sci. Stat. Comput., 3:76–110, 1982.
P. Colella. A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Stat. Comput., 6:104–117, 1985.
P. Colella. Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys., 87:171–200, 1990.
P. Colella and H. M. Glaz. Efficient solution algorithms for the Riemann problem for real gases. J. Comput. Phys., 59:264–284, 1985.
P. Colella, A. Majda, and Y. Roytburd. Theoretical and numerical structure for reacting shock waves. SIAM J. Sci. Stat. Comput., 7:1059–1080, 1986.
P. Colella and P. Woodward. The piecewise-parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54:174–201, 1984.
R. Courant and K. O. Friedrichs. Supersonic Flow and Shock Waves. Springer, 1948.
R. Courant, K. O. Friedrichs, and H. Lewy. über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., 100:32–74, 1928.
T. G. Cowling. Magnetohydrodynamics. Adam Hilger, The Institute of Physics, Bristol, England, 1976.
W. Dai and P. R. Woodward. On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamical flows. preprint.
W. Dai and P. R. Woodward. An approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys., 111:354–372, 1994.
W. Dai and P. R. Woodward. A high-order Godunov-type scheme for shock interactions in ideal magnetohydrodynamics. SIAM J. Sci. Comput., 18:957–981, 1997.
W. Dai and P. R. Woodward. An iterative Riemann solver for relativistic hydrodynamics. SIAM J. Sci. Comput., 18:982–995, 1997.
M. B. Davies, M. Ruffert, W. Benz, and E. Müller. A comparison between SPH and PPM: simulations of stellar collisions. Astron. Astrophys., 272:430–441, 1993.
D. L. De Zeeuw, A. F. Nagy, T. I. Gombosi, K. G. Powell, and J. G. Luhmann. A new axisymmetric MHD model of the interaction of the solar wind with Venus. J. Geophys. Res., 101:4547–4556, 1996.
R. DeVore. Transport techniques for multidimensional compressible magnetohydrodynamics. J. Comput. Phys., 92:142, 1991.
Ruth Dgani, Rolf Walder, and Harry Nussbaumer. Stability analysis of colliding winds in a double star system. Astron. Astrophys., 278:209–225, 1993.
W. G. Dixon. Special Relativity. Cambridge University Press, Cambridge, 1978.
R. Donat. Studies on error propagation for certain nonlinear approximations to hyperbolic equations: discontinuities in derivatives. SIAM J. Numer. Anal., 31:655–679, 1994.
R. Donat and A. Marquina. Capturing shock reflections: an improved flux formula. J. Comput. Phys., 125:42–58, 1996.
E. A. Dorfi and M. U. Feuchtinger. Adaptive radiation hydrodynamics of pulsating stars. Computer Phys. Comm., 89:69–90, 1995.
G. C. Duncan and P. A. Hughes. Simulations of relativistic extragalactic jets. Astrophysical J., 436:L119–L122, 1994.
V. Dwarkadas, R. Chevalier, and J. Blondin. The shaping of planetary nebulae: Asymmetry in the external wind. Astrophys. J., 457:773, 1996.
Carl Eckart. The thermodynamics of irreversible processes. Phys. Rev., 58:919–924, 1940.
B. Einfeldt. On Godunov-type methods for gas dynamics. SIAM J. Num. Anal., 25:294–318, 1988.
B. Einfeldt, C. D. Munz, P. L. Roe, and B. Sjogreen. On Godunov type methods near low densities. J. Comput. Phys., 92:273–295, 1991.
P. G. Eltgroth. Similarity analysis for relativistic flow in one dimension. Phys. Fluids, 14:2631, 1971.
B. Engquist and S. Osher. Stable and entropy satisfying approximations for transonic flow calculations. Math. Comp., 34:45–75, 1980.
R. Eulderink and G. Mellema. General relativistic hydrodynamics with a Roe solver. Astron. Astrophys. Suppl. Ser., 110:587–623, 1995.
T. E. Faber. Fluid Dynamics for Physicists. Cambridge University Press, Cambridge, 1995.
J. Falcovitz and M. Ben-Artzi. Recent developments of the GRP method. JSME International J. Series B, 38:497–517, 1995.
M. Fey, R. Jeltsch, and A.-T. Morel. Multidimensional schemes for nonlinear systems of hyperbolic conservation laws. In D. F. Griffiths and G. A. Watson, editors, 16th Biannual Dundee Conference on Numerical Analysis, 1995. Longman, 1996.
C. A. J. Fletcher. Computational Techniques for Fluid Dynamics. Springer-Verlag, 1988.
J. A. Font, J. Ma. Ibäñez, A. Marquina, and J. Ma. Martc\aci. Multidimensional relativistic hydrodynamics: characteristic fields and modern high-resolution shock-capturing schemes. Astron. Astrophys, 282:304–314, 1994.
J. Foster and J. D. Nightingale. A Short Course in General Relativity. Springer-Verlag, New York, 1995.
A. Frank, T. W. Jones, and D. Ryu. Time dependent simulation of oblique MHD cosmic-ray shocks using the two-fluid model. Astrophys. J., 441:629, 1995.
A. Frank and G. Mellema. From the owl to the eskimo: Radiation gasdynamics of planetary nebulae-IV. Astrophys. J., 430:800, 1994.
H. Freistühler. Some remarks on the structure of intermediate magnetohydrodynamic shocks. J. Geophys. Res., 96:3825–3827, 1991.
H. Freistühler and E. B. Pitman. A numerical study of a rotationally degenerate hyperbolic system, Part I: the Riemann problem. J. Comput. Phys., 100:306, 1992.
H. Friedrich. Hyperbolic reductions for Einstein’s equations. Class. Quantum Grav., 13:1451–1469, 1996.
Fryxell, E. Müller, and Arnett. Hydrodynamics and nuclear burning. MPA Report 449, 1989.
H. M. Glaz and T.-P. Liu. The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow. Advances Appl. Math., 5:111–146, 1984.
J. Glimm, G. Marshall, and B. Plohr. A generalized Riemann problem for quasi-one-dimensional gas flows. Advances Appl. Math., 5:1–30, 1984.
E. Godlewski and P.-A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York, 1996.
S. K. Godunov. Mat. Sb., 47:271, 1959.
T. I Gombosi, K. G. Powell, and D. L. De Zeeuw. Axisymmetric modelling of cometary mass loading on an adaptively refined grid. J. Geophys. Res., 99,A11: 21525–21539, 1994.
J. B. Goodman and R. J. LeVeque. A geometric approach to high resolution TVD schemes. SIAM J. Num. Anal., 25:268–284, 1988.
A. Greenbaum. Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, 1997.
F. H. Harlow and J. E. Welch. The MAC method: a computing technique for solving viscous, incompressible, transient fluid flow problems involving free surfaces. Phys. Fluids, 8:2182–2189, 1965.
A. Harten. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49:357–393, 1983.
A. Harten, B. Engquist, S. Osher, and S. Chakravarthy. Uniformly high order accurate essentially nonoscillatory schemes, III. J. Comput. Phys., 71:231, 1987.
A. Harten and J. M. Hyman. Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys., 50:235–269, 1983.
A. Harten, J. M. Hyman, and P. D. Lax. On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math., 29:297–322 (with appendix by Barbara Keyfitz), 1976.
A. Harten and P. D. Lax. A random choice finite difference scheme for hyperbolic conservation laws. SIAM J. Num. Anal., 18:289–315, 1981.
A. Harten, P. D. Lax, and B. van Leer. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25:35–61, 1983.
A. Harten and S. Osher. Uniformly high-order accurate nonoscillatory schemes. I. SIAM J. Num. Anal., 24:279–309, 1987.
A. Harten and G. Zwas. Self-adjusting hybrid schemes for shock computations. J. Comput. Phys., 9:568, 1972.
C. Hirsch. Numerical Computation of Internal and External Flows. Wiley, 1988.
H. T. Huynh. Accurate upwind methods for the Euler equations. SIAM J. Numer. Anal., 32:1565–1619, 1995.
V. Icke, B. Balick, and A. Frank. The hydrodynamics of aspherical planetary nebulae. Astron. Astrophys., 253:224–243, 1991.
E. Isaacson, D. Marchesin, and B. Plohr. Transitional waves for conservation laws. SIAM J. Math. Anal., 21:837–866, 1990.
A. Iserles. Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge, 1996.
W. Israel. Relativistic theory of shock waves. Proc. Roy. Soc. London, A259:129, 1960.
J. D. Jackson. Classical Electrodynamics. John Wiley and Sons, Inc., New York, 1975.
A. Jeffrey and T. Taniuti. Non-linear Wave Propagation. Academic Press, 1964.
G. Jiang and C. W. Shu. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126:202–228, 1996.
S. Jin. Implicit numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys., to appear.
S. Jin and J.-G. Liu. The effects of numerical viscosities: I. Slowly moving shocks. J. Comput. Phys., 126:373–389, 1996.
Montgomery H. Johnson and Christopher F. McKee. Relativistic hydrodynamics in one dimension. Phys. Rev. D, 3:858–863, 1971.
O. S. Jones, U. Shumlak, and D. S. Eberhardt. An implicit scheme for nonideal magnetohydrodynamics. J. Comput. Phys., 130:231–242, 1997.
A. Kantrowitz and H. E. Petschek. MHD characteristics and shock waves. In W. B. Kunkel, editor, Plasma Physics in Theory and Application, pages 148–206. McGraw-Hill, 1966.
S. Karni and S. Canic. Computation of slowly moving shocks. J. Comput. Phys., to appear, 1997.
J. Kevorkian. Partial Differential Equations. Wadsworth & Brooks/Cole, 1990.
B. L. Keyfitz and H. C. Kranzer. A system of hyperbolic conservation laws arising in elasticity theory. Arch. Rat. Mech. Anal., 72:219–241, 1980.
Arieh Königl. Relativistic gasdynamics in two dimensions. Phys. Fluids, 23:1083, 1980.
N. A. Krall and A. W. Trievelpiece. Principles of Plasma Physics. McGraw-Hill, New York, 1973.
D. Kröner. Numerical Schemes for Conservation Laws. Wiley-Teubner series, 1997.
J. D. Lambert. Computational Methods in Ordinary Differential Equations. Wiley, 1973.
L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Pergamon Press Inc., Elmsford, New York, 1979.
J. O. Langseth and R. J. LeVeque. Three-dimensional Euler computations using CLAWPACK. In P. Arminjon, editor, Conf. on Numer. Meth. for Euler and Navier-Stokes Eq., Montreal, 1995. to appear.
J. O. Langseth and R. J. LeVeque. A wave-propagation method for three-dimensional hyperbolic conservation laws. preprint, 1997.
P. D. Lax. Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math., 10:537–566, 1957.
P. D. Lax. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM Regional Conference Series in Applied Mathematics, #11, 1972.
B. P. Leonard, A. P. Lock, and M. K. MacVean. Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes. Monthly Weather Rev., 124:2588–2606, 1996.
R. J. LeVeque. CLAWPACK software. available from netlib.att.com in netlib/pdes/claw or on the Web at the URL http://www.amath.washington.edu/~rj1/clawpack.html.
R. J. LeVeque. Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations. Math. Comput., 47:37–54, 1986.
R. J. LeVeque. Hyperbolic conservation laws and numerical methods. Von Karman Institute for Fluid Dynamics Lecture Series, 90–03, 1990.
R. J. LeVeque. Numerical Methods for Conservation Laws. Birkhäuser-Verlag, 1990.
R. J. LeVeque. Finite difference methods for differential equations. Class Notes, http://www.amath.washington.edu/~rjl, 1996.
R. J. LeVeque. Balancing source terms and flux gradients in high-resolution Godunov methods. in preparation, 1997.
R. J. LeVeque. Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys., 131:327–353, 1997.
R. J. LeVeque and K.-M. Shyue. One-dimensional front tracking based on high resolution wave propagation methods. SIAM J. Sci. Comput., 16:348–377, 1995.
R. J. LeVeque and K.-M. Shyue. Two-dimensional front tracking based on high resolution wave propagation methods. J. Comput. Phys., 123:354–368, 1996.
R. J. LeVeque and R. Walder. Grid alignment effects and rotated methods for computing complex flows in astrophysics. GAMM Conf. on Comput. Fluid Dyn., Lausanne, 1991.
R. J. LeVeque and H. C. Yee. A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys., 86:187–210, 1990.
E. P. T. Liang. Relativistic simple waves: Shock damping and entropy production. Astrophys. J., 211:361–376, 1977.
A. Lichnerowicz. Relativistic Hydrodynamics and Magnetohydrodynamics. Benjamin, New York, 1967.
I. Lindemuth and J. Killeen. Alternating direction implicit techniques for two-dimensional magnetohydrodynamic calculations. J. Comput. Phys., 13:181, 1973.
M.-S. Liou. A sequel to AUSM: AUSM+. J. Comput. Phys., to appear.
M.-S. Liou and C. J. Steffen Jr. A new flux splitting scheme. J. Comput. Phys., 107:107, 1993.
T. P. Liu. On the viscosity criterion for hyperbolic conservation laws. In M. Shearer, editor, Viscous Profiles and Numerical Methods for Shock Waves, page 105, Philadelphia, 1991. SIAM.
X.-D. Liu, S. Osher, and T. Chan. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115:200–212, 1994.
M.-M. Mac Low and M. Normann. Nonlinear growth of dynamical overstabilities in blast waves. Astrophys. J., 407:207–218, 1993.
A. Marquina. Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws. SIAM J. Sci. Comput., 15:892, 1994.
A. Marquina, J. Ma. MartÃ, J. Ma. Ibáañez, J. A. Miralles, and R. Donat. Ultra-relativistic hydrodynamics: high-resolution shock-capturing methods. Astron. Astrophys., 258:566–571, 1992.
R. L. Marsa and M. W. Choptuik. Black hole-scalar field interactions in spherical symmetry, preprint, 1996.
J. M. Martà and E. Müller. The analytical solution of the Riemann problem in relativistic hydrodynamics. J. Fluid Mech., 258:317–333, 1994.
J. Ma. MartÃ, J. Ma. Ibáñez, and J. A. Miralles. Numerical relativistic hydro-dynamics: Local characteristic approach. Phys. Rev. D, 43:3794–3801, 1991.
J. Ma. Martà and E. Müller. Extension of the piecewise parabolic method to one-dimensional relativistic hydrodynamics. J. Comput. Phys., 123:1–14, 1996.
Joan Massó, Edward Seidel, and Paul Walker. Adaptive mesh refinement in numerical relativity. To appear in General Relativity(MG7 Proceedings), R. Ruffini and M. Keiser (eds.), World Scientific, 1995.
G. Mellema, F. Eulderink, and V. Icke. Hydrodynamical models of aspherical planetary nebulae. Astron. Astrophys., 252:718–732, 1991.
L. Mestel and N. O. Weiss. Magnetohydrodynamics. Fourth Saas-Fee Course, 1974.
D. Mihalas and B. W. Mihalas. Foundations of Radiation Hydrodynamics. Oxford Univesity Press, 1984.
C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. W. H. Freeman and Co., New York, 1973.
C. Moller. The Theory of Relativity, 2nd ed. Oxford, University Press, Oxford, 1972.
K. W. Morton and D. F. Mayers. Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge, 1994.
A. Mukherjee. An adaptive finite element code for elliptic boundary value problems in three dimensions with applications in numerical relativity. PhD thesis, Mathematics, Pennsylvania State University, 1996.
W. A. Mulder. Computation of a quasi-steady gas flow in a spiral galaxy by means of a multigrid method. Astron. Astrophys., 156:354–380, 1986.
E. Müller and Steinmetz. Simulating self-gravitating hydrodynamics. Comp. Phys. Comm., 89:45–58, 1995.
R. S. Myong. Theoretical and computational investigations of nonlinear waves in magnetohydrodynamics. PhD thesis, Aerospace Engineering, University of Michigan, 1996.
Gregory L. Naber. The Geometry of Minkowski Spacetime. Springer-Verlag, New York, 1992.
W. F. Noh. Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J. Comput. Phys., 72:78, 1987.
M. L. Norman and K-H.A. Winkler. Astrophysical Radiation Hydrodynamics. Reidel, Norwell, MA, 1986.
H. Nussbaumer and R. Walder. Modification of the nebular environment in symbiotic systems due to colliding winds. Astronomy and Astrophysics, 278:209–225, 1993.
E. S. Oran and J. P. Boris. Numerical Simulation of Reactive Flow. Elsevier, 1987.
S. Osher. Riemann solvers, the entropy condition, and difference approximations. SIAM J. Num. Anal., 21:217–235, 1984.
S. Osher and S. Chakravarthy. High resolution schemes and the entropy condition. SIAM J. Num. Anal., 21:995–984, 1984.
S. Osher and F. Solomon. Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comp., 38:339–374, 1982.
W. Pauli. Theory of Relativity. Dover Publications, Inc., New York, 1981.
R. B. Pember. Numerical methods for hyperbolic conservation laws with stiff relaxation, I. Spurious solutions. SIAM J. Appl. Math., 53:1293–1330, 1993.
R. B. Pember. Numerical methods for hyperbolic conservation laws with stiff relaxation, II. Higher order Godunov methods. SIAM J. Sci. Comput., 14, 1993.
R. Peyret and T. D. Taylor. Computational Methods for Fluid Flow. Springer, 1983.
K. Powell. Solution of the Euler and Magnetohydrodynamic Equations on Solution-Adaptive Cartesian Grids. Von Karman Institute for Fluid Dynamics Lecture Series, 1996.
K. G. Powell. An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). ICASE Report No. 94-24, NASA Langley Research Center, 1994.
K. G. Powell, P. L. Roe, R. S. Myong, T. Gombosi, and D. De Zeeuw. An upwind scheme for magnetohydrodynamics. AIAA-95-1704-CP, 1995.
K. H. Prendergast and K. Xu. Numerical hydrodynamics from gas-kinetic theory. J. Comput. Phys., 109:53–66, 1993.
J. J. Quirk. A contribution to the great Riemann solver debate. Internat. J. Numer. Methods Fluids, 18:555–574, 1994.
J. Ramshaw. A method for enforcing the solenoidal condition on magnetic fields in numerical calculations. J. Comput. Phys., 52:592–596, 1983.
R. D. Richtmyer and K. W. Morton. Difference Methods for Initial-value Problems. Wiley-Interscience, 1967.
T. W. Roberts. The behavior of flux difference splitting schemes near slowly moving shock waves. J. Comput. Phys., 90:141–160, 1990.
P. Roe. The Harten memorial lecture — new applications of upwinding. preprint, 1997.
P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43:357–372, 1981.
P. L. Roe. Numerical algorithms for the linear wave equation. Royal Aircraft Establishment Technical Report 81047, 1981.
P. L. Roe. Some contributions to the modeling of discontinuous flows. Lect. Notes Appl. Math., 22:163–193, 1985.
P. L. Roe. Upwind differencing schemes for hyperbolic conservation laws with source terms. In C. Carraso, P.-A. Raviart, and D. Serre, editors, Nonlinear Hyperbolic Problems, pages 41–51. Springer-Verlag, Lecture Notes in Mathematics 1270, 1986.
P. L. Roe and D. S. Balsara. Notes on the eigensystem of magnetohydrodynamics. SIAM J. Appl. Math., 1996.
P. L. Roe and D. Sidilkover. Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Num. Anal., 29:1542–1568, 1992.
J. Romero, J. Ma. Ibáñez, J. Ma Marti, and J. A. Miralles. A new spherically symmetric general relativistic hydrodynamical code. Astrophys. J., 462:836, 1996.
D. Ryu and T. W. Jones. Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow. Astrophysical J., 442:228–258, 1995.
D. Ryu, T. W. Jones, and A. Frank. Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for multi-dimensional flow. Astrophysical J., 452:785–796, 1995.
D. Ryu and E.T. Vishniac. The dynamic instability of adiabatic blast waves. Astrophys. J., 368:411–425, 1991.
J. Saltzman. An unsplit 3-D upwind method for hyperbolic conservation laws. J. Comput. Phys., pages 153–168, 1994.
R. H. Sanders and K. H. Prendergast. The possible relation of the 3-kiloparsec arm to explosions in the galactic nucleus. Astrophys. J., 188:489, 1974.
D. G. Schaeffer and M. Shearer. The classification of 2 x 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery. Comm. Pure Appl. Math., 40:141–178, 1987.
D. G. Schaeffer and M. Shearer. Riemann problems for nonstrictly hyperbolic 2 x 2 systems of conservation laws. Trans. Amer. Math. Soc., 304:267–306, 1987.
S. Schecter and M. Shearer. Transversality for undercompressive shocks in Riemann problems. In M. Shearer, editor, Viscous Profiles and Numerical Methods for Shock Waves, pages 142–154, Philadelphia, 1991. SIAM.
D. Schnack and J. Killeen. Nonlinear, two-dimensional magnetohydrodynamic calculations. J. Comput. Phys., 35:110, 1980.
Bernard F. Schutz. A first Course in General Relativity. Cambridge University Press, Cambridge, 1990.
E. Seidel and W.-M. Suen. Numerical relativity, preprint.
E. Seidel and W.-M. Suen. Towards a singularity proof in numerical relativity. Phys. Rev. Let., 69:1845–1848, 1992.
C.-W. Shu. TVB uniformly high-order schemes for conservation laws. Math. Comp., 49:105–121, 1987.
C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys., 77:439–471, 1988.
C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys., 83:32, 1989.
F. H. Shu. The Physics of Astrophysics: Volume 1, Radiation. University Science Books, Mill Valley, CA, 1991.
F. H. Shu. The Physics of Astrophysics: Volume 2, Gas Dynamics. University Science Books, Mill Valley, CA, 1991.
D. Sidilkover. A genuinely multidimensional upwind scheme for the compressible Euler equations. In J. Glimm et al., editors, Proceedings of the Fifth International Conference on Hyperbolic Problems: Theory, Numerics, Applications. World Scientific, June 1994.
P. K. Smolarkeiwicz and W. W. Grabowski. The multidimensional positive definite advection transport algorithm: nonoscillatory option. J. Comput. Phys., 86:355–375, 1990.
J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer, 1983.
J. Smoller and B. Temple. Global solutions of the relativistic Euler equations. Comm. Math. Phys., 156:67–99, 1993.
J. Smoller and B. Temple. Shock-waves and irreversibility in Einstein’s theory of gravity. In J. Glimm et al., editors, Proceedings of the Fifth International Conference on Hyperbolic Problems: Theory, Numerics, Applications, pages 81–90, Singapore, June 1994. World Scientific.
J. Smoller and B. Temple. An astrophysical shock-wave solution of the Einstein equations. Phys. Rev. D, 51:2733–2743, 1996.
G. Sod. A survey of several finite diffeence methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys., 27:1–31, 1978.
G. Sod. Numerical Methods for Fluid Dynamics. Cambridge University Press, 1985.
J. L. Steger and R. F. Warming. Flux vector splitting of the inviscid gasdynamic equations with applications to finite-difference methods. J. Comput. Phys., 40:263, 1981.
O. Steiner, M. Knölker, and M. Schüssler. Dynamic interaction of convection with magnetic flux sheets: first results of a new MHD code. In R. J. Rutten and C. J. Schrijver, editors, Solar Surface Magnetism, NATO Advanced Research Workshop, Dordrecht, 1993. Kluwer Academic Publishers.
R. S. Steinolfson and A. J. Hundhausen. MHD intermediate shocks in coronal mass ejections. J. Geophys. Res., 95:6389, 1990.
I. R. Stevens, J. M. Blondin, and A. M. T. Pollock. Colliding winds from early-type stars in binary systems. Ap.J., 386:265–287, 1992.
G. Strang. On the construction and comparison of difference schemes. SIAM J. Num. Anal., 5:506–517, 1968.
J. C. Strikwerda. Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks/Cole, 1989.
Paul N. Swarztrauber. Fast Poisson solvers. In Gene H. Golub, editor, Studies in Numerical Analysis, volume 24, pages 319–370, Washington, D.C., 1984. The Mathematical Association of America.
P. K. Sweby. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal., 21:995–1011, 1984.
J. L. Synge. The Relativistic Gas. North-Holland Publishing Company, Amsterdam, 1957.
A. H. Taub. Relativistic Rankine-Hugoniot equations. Phys. Rev., 74:328, 1948.
Kevin W. Thompson. The special relativistic shock tube. J. Fluid Mech., 171:365–375, 1986.
K. S. Thorne. Gravitational radiation. In S. W. Hawking and W. Israel, editors, 300 Years of Gravitation. Cambridge University Press, 1987.
K. S. Thorne. Black Holes and Time Warps. W. W. Norton & Co., New York, 1994.
K. S. Thorne. Gravitational waves. In E. W. Kolb and R. D. Peccei, editors, Particle and Nuclear Astrophysics and Cosmology in the Next Millennium, Proc. 1994 Snowmass Summer School, pages 160–184. World Scientific, 1994.
K.S. Thorne. Relativistic shock: The Taub adiabat. Astrophys. J., 179:897, 1973.
E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag, Berlin, Heidelberg, 1997.
J. A. Trangenstein. An unsplit Godunov method for three-dimensional polymer flooding. preprint, 1993.
L. N. Trefethen and D. Bau. Numerical Linear Algebra. SIAM, Philadelphia, 1997.
G. D. van Albada, B. van Leer, and W. W. Roberts, Jr. A comparative study of computational methods in cosmic gas dynamics. Astron. Astrophys., 108:76–84, 1982.
B. van Leer. Towards the ultimate conservative difference scheme I. The quest of monotonicity. Springer Lecture Notes in Physics, 18:163–168, 1973.
B. van Leer. Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Comput. Phys., 23:276–299, 1977.
B. van Leer. Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J. Comput. Phys., 32:101–136, 1979.
B. van Leer. Flux-vector splitting for the Euler equations. In E. Krause, editor, Lecture Notes in Physics, volume 170, page 507. Springer-Verlag, 1982.
B. van Leer. On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher, and Roe. SIAM J. Sci. Stat. Comput., 5:1–20, 1984.
E. T. Vishniac. Nonlinear instabilities in shock-bounded slabs. Astrophys. J., 428:186–208, 1994.
R. M. Wald. General Relativity. Univesity of Chicago Press, 1984.
R. Walder. Some Aspects of the Computational Dynamics of Colliding Flows in Astrophysical Nebulae. PhD thesis, Astronomy Institute, ETH-Zürich, No. 10302, 1993.
R. Walder and D. Folini. Radiative cooling instability in ID colliding flows. Astronomy and Astrophysics, 315:265–284, 1996.
P. Wesseling. An Introduction to Multigrid Methods. Wiley, New York, 1992.
G. Whitham. Linear and Nonlinear Waves. Wiley-Interscience, 1974.
F. A. Williams. Combustion Theory. Benjamin/Cummings, 1985.
P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54:115–173, 1984.
C. C. Wu. On MHD intermediate shocks. Geophys. Res. Letters, 14:668–671, 1987.
C. C. Wu. Formation, structure, and stability of MHD intermediate shocks. J. Geophys. Res., 95:8149, 1990.
C. C. Wu. New theory of MHD shock waves. In M. Shearer, editor, Viscous Profiles and Numerical Methods for Shock Waves, pages 231–235, Philadelphia, 1991. SIAM.
C. C. Wu and C. F. Kennel. Evolution of small amplitude intermediate shocks in a dissipative and dispersive system. J. Plasma Phys., 47:85, 1992.
H. C. Yee. Upwind and symmetric shock-capturing schemes. NASA Ames Technical Memorandum 89464, 1987.
A. L. Zachary and P. Colella. A higher order Godunov method for the equations of ideal magnetohydrodynamics. J. Comput. Phys., 99:341–347, 1992.
A. L. Zachary, A. Malagoli, and P. Colella. A higher-order Godunov method for multidimensional ideal magnetohydrodynamics. SIAM J. Sci. Comput., 15:263–284, 1994.
S. T. Zalesak. Fully multidimensional flux corrected transport algorithms for fluids. J. Comput. Phys., 31:335–362, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
LeVeque, R.J. (1998). Nonlinear Conservation Laws and Finite Volume Methods. In: Steiner, O., Gautschy, A. (eds) Computational Methods for Astrophysical Fluid Flow. Saas-Fee Advanced Courses, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31632-9_1
Download citation
DOI: https://doi.org/10.1007/3-540-31632-9_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64448-4
Online ISBN: 978-3-540-31632-9
eBook Packages: Springer Book Archive