- 736 Downloads
The appropriate physical balance laws are derived for m-physical quantities, u1,...,um with u = t(u1,...,Um) and u(x,t) defined for x = (x1,...,xN) ∈ RN (N = 1,2, or 3), t ≥ 0 and with the values u(x,t) lying in an open subset, G, of Rm, the state space. The state space G arises because physical quantities such as the density or total energy should always be positive; thus the values of u are often constrained to an open set G.
The flux functions appearing in these balance laws are through prescribed nonlinear functions, Fj(u), mapping G into Rm, j = 1,...,N while source terms are defined by S(u,x,t) with S a given smooth function of these arguments with values in Rm. In particular, the detailed microscopic effects of diffusion and dissipation are ignored.
A generalized version of the principle of virtual work is applied (see Antman ).
KeywordsNonlinear Wave Equation Entropy Condition Simple Wave Shallow Water Theory Nonlinear Wave Propagation
Unable to display preview. Download preview PDF.
Bibliography for Chapter 1
- Courant, R., and K. O. Friedrichs: Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1949.Google Scholar
- Federer, H.: Geometric Measure Theory, Springer-Verlag, New York, 1968.Google Scholar
- Ficket, W., and W. C. Davis: Detonation, University of California Press, Berkeley, 1979.Google Scholar
- Gardner, R. A.: “On the detonation of a combustible gas”, preprint, 1981.Google Scholar
- Harten, A.: “On the symmetric form of systems of conservation laws with entropy”, I.C.A.S.E. Report No. 81-34, October 1981.Google Scholar
- Hunter, J.: “Weakly nonlinear wave propagation, Ph.D. Thesis, Stanford University, 1981.Google Scholar
- Hunter, J.: “Weakly nonlinear high frequency waves”, M.R.C. Report #2381, May 1982.Google Scholar
- Kevorkian, J., and Cole, J. D.: Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1980.Google Scholar
- Landau, L. D., and E. M. Lifschitz: Fluid Mechanics, Addison Wesley, Reading, MA, 1971.Google Scholar
- Lax, P. D.: “Hyperbolic systems of conservation laws and the mathematical theory of shock waves”, Regional Conf. Series in Appl. Math., #13, SIAM, 1973.Google Scholar
- Majda, A., and R. Pego: Stable viscosity matrices and conservation laws”, (submitted to J. Differential Equations).Google Scholar
- Majda, A., and R. Rosales: “A theory for Mach stem formation in reacting shock fronts: I, the basic perturbation analysis”, (to appear in 1983 in SIAM J. Appl. Math.).Google Scholar
- Majda, A., and R. Rosales: “A theory for Mach stem formation in reacting shock fronts: II, the evidence for breakdown”, (in preparation).Google Scholar
- Mock, M. S.: “Systems of conservation laws of mixed type”, (to appear in J. Differential Equations).Google Scholar
- Nayfeh, A. H.: “A comparison of perturbation methods for nonlinear hyperbolic waves” in Singular Perturbations and Asymptotics, edited by R. Meyer and S. Partner, 1980, 223–276.Google Scholar
- Pego, R.: “Viscosity matrices for systems of conservation laws”, Ph.D. thesis, University of California, Berkeley, 1982.Google Scholar
- Rosales, R., and A. Majda: “Weakly nonlinear detonation waves” (to appear in SIAM J. Appl. Math. in 1983).Google Scholar
- Roytburd, U.: “Hopf bifurcation in a model for solid fuel combustion”, Ph.D. Thesis, University of California, Berkeley, 1981.Google Scholar
- Serrin, J.: Mathematical Principles of Classical Fluid Mechanics, in Handbuch der Physik, 8, Springer-Verlag, 1959.Google Scholar
- Strehlow, R. A.: Fundamentals of Combustion, Krieger Publishing, New York, revised edition, 1979.Google Scholar
- Williams, F. A.: Combustion Theory, Addison-Wesley, Reading, MA, 1964.Google Scholar
Some References for Numerical Solution of Conservation Laws
- [N-3]Colella, P., and P. Woodward: “The numerical simulation of two-dimensional fluid flow with strong shocks”, to appear in J. Comp. Phys.Google Scholar
- [N-7]Harten, A.: “High resolution schemes for hyperbolic conservation laws”, to appear in J. Comp. Phys.Google Scholar
- [N-9]Harten, A., Lax, P. D., and B. Van Leer: “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws”, I.C.A.S.E. Report #82-5, March 1982 (to appear in SIAM Review).Google Scholar