846 Chapter 19.Partial Differential Equations In summary,our recommendation for initial value problems that can be cast in flux-conservative form,and especially problems related to the wave equation,is to use the staggered leapfrog method when possible.We have personally had better success with it than with the Two-Step Lax-Wendroff method.For problems sensitive to transport errors,upwind differencing or one of its refinements should be considered. Fluid Dynamics with Shocks As we alluded to earlier,the treatment of fluid dynamics problems with shocks has become a very complicated and very sophisticated subject.All we can attempt 8 to do here is to guide you to some starting points in the literature. There are basically three important general methods for handling shocks.The g oldest and simplest method,invented by von Neumann and Richtmyer,is to add artificial viscosity to the equations,modeling the way Nature uses real viscosity to smooth discontinuities.A good starting point for trying out this method is the differencing scheme in $12.11 of [11.This scheme is excellent for nearly all problems in one spatial dimension. The second method combines a high-order differencing scheme that is accurate for smooth flows with a low order scheme that is very dissipative and can smooth 9 the shocks.Typically,various upwind differencing schemes are combined using weights chosen to zero the low order scheme unless steep gradients are present,and also chosen to enforce various"monotonicity"constraints that prevent nonphysical 9 oscillations from appearing in the numerical solution.References [2-3,5]are a good place to start with these methods. 令芝g The third,and potentially most powerful method,is Godunov's approach.Here one gives up the simple linearization inherent in finite differencing based on Taylor series and includes the nonlinearity of the equations explicitly.There is an analytic 6 solution for the evolution of two uniform states of a fluid separated by a discontinuity, the Riemann shock problem.Godunov's idea was to approximate the fluid by a large number of cells of uniform states,and piece them together using the Riemann solution.There have been many generalizations of Godunov's approach,of which the most powerful is probably the PPM method [6]. Readable reviews of all these methods,discussing the difficulties arising when 9em Numerica 10621 one-dimensional methods are generalized to multidimensions,are given in [7-91. CITED REFERENCES AND FURTHER READING: Ames.W.F.1977.Numerical Methods for Partial Differential Equations,2nd ed.(New York: Academic Press),Chapter 4. Richtmyer,R.D.,and Morton,K.W.1967.Difference Methods for Initial Value Problems,2nd ed. (New York:Wiley-Interscience).[1] Centrella,J.,and Wilson,J.R.1984,Astrophysical Journal Supplement,vol.54,pp.229-249, Appendix B.[2] Hawley,J.F.,Smarr,L.L.,and Wilson,J.R.1984,Astrophysical Journal Supplement,vol.55, pp.211-246,82c.[3] Kreiss,H.-O.1978,Numerical Methods for Solving Time-Dependent Problems for Partial Differ- ential Equations(Montreal:University of Montreal Press),pp.66ff.[4] Harten,A.,Lax,P.D.,and Van Leer,B.1983,S/AM Review,vol.25,pp.36-61.[5] Woodward,P.,and Colella,P.1984,Journal of Computationa/Physics,vol.54,pp.174-201.[6] Roache,P.J.1976,Computational Fluid Dynamics (Albuquerque:Hermosa).[7]846 Chapter 19. Partial Differential Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). In summary, our recommendation for initial value problems that can be cast in flux-conservativeform, and especially problems related to the wave equation, is to use the staggered leapfrog method when possible. We have personally had better success with it than with the Two-Step Lax-Wendroff method. For problems sensitive to transport errors, upwind differencing or one of its refinements should be considered. Fluid Dynamics with Shocks As we alluded to earlier, the treatment of fluid dynamics problems with shocks has become a very complicated and very sophisticated subject. All we can attempt to do here is to guide you to some starting points in the literature. There are basically three important general methods for handling shocks. The oldest and simplest method, invented by von Neumann and Richtmyer, is to add artificial viscosity to the equations, modeling the way Nature uses real viscosity to smooth discontinuities. A good starting point for trying out this method is the differencing scheme in §12.11 of [1]. This scheme is excellent for nearly all problems in one spatial dimension. The second method combines a high-order differencing scheme that is accurate for smooth flows with a low order scheme that is very dissipative and can smooth the shocks. Typically, various upwind differencing schemes are combined using weights chosen to zero the low order scheme unless steep gradients are present, and also chosen to enforce various “monotonicity” constraints that prevent nonphysical oscillations from appearing in the numerical solution. References [2-3,5] are a good place to start with these methods. The third, and potentially most powerful method, is Godunov’s approach. Here one gives up the simple linearization inherent in finite differencing based on Taylor series and includes the nonlinearity of the equations explicitly. There is an analytic solution for the evolution of two uniform states of a fluid separated by a discontinuity, the Riemann shock problem. Godunov’s idea was to approximate the fluid by a large number of cells of uniform states, and piece them together using the Riemann solution. There have been many generalizations of Godunov’s approach, of which the most powerful is probably the PPM method [6]. Readable reviews of all these methods, discussing the difficulties arising when one-dimensional methods are generalized to multidimensions, are given in [7-9]. CITED REFERENCES AND FURTHER READING: Ames, W.F. 1977, Numerical Methods for Partial Differential Equations, 2nd ed. (New York: Academic Press), Chapter 4. Richtmyer, R.D., and Morton, K.W. 1967, Difference Methods for Initial Value Problems, 2nd ed. (New York: Wiley-Interscience). [1] Centrella, J., and Wilson, J.R. 1984, Astrophysical Journal Supplement, vol. 54, pp. 229–249, Appendix B. [2] Hawley, J.F., Smarr, L.L., and Wilson, J.R. 1984, Astrophysical Journal Supplement, vol. 55, pp. 211–246, §2c. [3] Kreiss, H.-O. 1978, Numerical Methods for Solving Time-Dependent Problems for Partial Differ￾ential Equations (Montreal: University of Montreal Press), pp. 66ff. [4] Harten, A., Lax, P.D., and Van Leer, B. 1983, SIAM Review, vol. 25, pp. 36–61. [5] Woodward, P., and Colella, P. 1984, Journal of Computational Physics, vol. 54, pp. 174–201. [6] Roache, P.J. 1976, Computational Fluid Dynamics (Albuquerque: Hermosa). [7]