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834Chapter 19.Partial Differential Equationsengineering; these methods allow considerable freedom in putting computationalelements where you want them, important when dealing with highly irregular geometries. Spectral methods [13-15] are preferred for very regular geometries and smoothfunctions; they converge more rapidly than finite-difference methods (cf. §19.4), butthey do not work well for problems with discontinuities.Ames, W.F.
1977, Numerical Methods for Partial Differential Equations, 2nd ed. (New York:Academic Press). [1]Richtmyer, R.D., and Morton, K.W. 1967, Difference Methods for Initial Value Problems, 2nd ed.(New York: Wiley-Interscience). [2]Roache, P.J. 1976, Computational Fluid Dynamics (Albuquerque: Hermosa). [3]Mitchell, A.R., and Griffiths, D.F. 1980, The Finite Difference Method in Partial Differential Equations (New York: Wiley) [includes discussion of finite element methods]. [4]Dorr, F.W. 1970, SIAM Review, vol.
12, pp. 248–263. [5]Meijerink, J.A., and van der Vorst, H.A. 1977, Mathematics of Computation, vol. 31, pp. 148–162. [6]van der Vorst, H.A. 1981, Journal of Computational Physics, vol. 44, pp. 1–19 [review of sparseiterative methods]. [7]Kershaw, D.S. 1970, Journal of Computational Physics, vol. 26, pp. 43–65. [8]Stone, H.J. 1968, SIAM Journal on Numerical Analysis, vol. 5, pp. 530–558.
[9]Jesshope, C.R. 1979, Computer Physics Communications, vol. 17, pp. 383–391. [10]Strang, G., and Fix, G. 1973, An Analysis of the Finite Element Method (Englewood Cliffs, NJ:Prentice-Hall). [11]Burnett, D.S. 1987, Finite Element Analysis: From Concepts to Applications (Reading, MA:Addison-Wesley). [12]Gottlieb, D. and Orszag, S.A. 1977, Numerical Analysis of Spectral Methods: Theory and Applications (Philadelphia: S.I.A.M.).
[13]Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. 1988, Spectral Methods in FluidDynamics (New York: Springer-Verlag). [14]Boyd, J.P. 1989, Chebyshev and Fourier Spectral Methods (New York: Springer-Verlag). [15]19.1 Flux-Conservative Initial Value ProblemsA large class of initial value (time-evolution) PDEs in one space dimension canbe cast into the form of a flux-conservative equation,∂F(u)∂u=−∂t∂x(19.1.1)where u and F are vectors, and where (in some cases) F may depend not only on ubut also on spatial derivatives of u. The vector F is called the conserved flux.For example, the prototypical hyperbolic equation, the one-dimensional waveequation with constant velocity of propagation v2∂2u2∂ u=v∂t2∂x2(19.1.2)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).CITED REFERENCES AND FURTHER READING:19.1 Flux-Conservative Initial Value Problems835can be rewritten as a set of two first-order equations∂s∂r=v∂t∂x∂r∂s=v∂t∂x(19.1.3)∂u∂x∂us≡∂tr≡v(19.1.4)In this case r and s become the two components of u, and the flux is given bythe linear matrix relation0 −vF(u) =·u(19.1.5)−v 0(The physicist-reader may recognize equations (19.1.3) as analogous to Maxwell’sequations for one-dimensional propagation of electromagnetic waves.)We will consider, in this section, a prototypical example of the general fluxconservative equation (19.1.1), namely the equation for a scalar u,∂u∂u= −v∂t∂x(19.1.6)with v a constant.
As it happens, we already know analytically that the generalsolution of this equation is a wave propagating in the positive x-direction,u = f(x − vt)(19.1.7)where f is an arbitrary function. However, the numerical strategies that we developwill be equally applicable to the more general equations represented by (19.1.1). Insome contexts, equation (19.1.6) is called an advective equation, because the quantityu is transported by a “fluid flow” with a velocity v.How do we go about finite differencing equation (19.1.6) (or, analogously,19.1.1)? The straightforward approach is to choose equally spaced points along boththe t- and x-axes.
Thus denotexj = x0 + j∆x,j = 0, 1, . . . , Jtn = t0 + n∆t,n = 0, 1, . . ., N(19.1.8)Let unj denote u(tn , xj ). We have several choices for representing the timederivative term. The obvious way is to setun+1− unj∂u j+ O(∆t)=∂t j,n∆t(19.1.9)This is called forward Euler differencing (cf. equation 16.1.1). While forward Euleris only first-order accurate in ∆t, it has the advantage that one is able to calculateSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.
To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).where836Chapter 19.Partial Differential EquationsFTCSt or nFigure 19.1.1. Representation of the Forward Time Centered Space (FTCS) differencing scheme.
In thisand subsequent figures, the open circle is the new point at which the solution is desired; filled circles areknown points whose function values are used in calculating the new point; the solid lines connect pointsthat are used to calculate spatial derivatives; the dashed lines connect points that are used to calculate timederivatives. The FTCS scheme is generally unstable for hyperbolic problems and cannot usually be used.quantities at timestep n + 1 in terms of only quantities known at timestep n. For thespace derivative, we can use a second-order representation still using only quantitiesknown at timestep n:unj+1 − unj−1∂u + O(∆x2 )=(19.1.10)∂x j,n2∆xThe resulting finite-difference approximation to equation (19.1.6) is called the FTCSrepresentation (Forward Time Centered Space), nun+1− unjuj+1 − unj−1j= −v(19.1.11)∆t2∆xwhich can easily be rearranged to be a formula for un+1in terms of the otherjquantities.
The FTCS scheme is illustrated in Figure 19.1.1. It’s a fine example ofan algorithm that is easy to derive, takes little storage, and executes quickly. Toobad it doesn’t work! (See below.)for eachThe FTCS representation is an explicit scheme. This means that un+1jj can be calculated explicitly from the quantities that are already known. Later weshall meet implicit schemes, which require us to solve implicit equations couplingthe un+1for various j. (Explicit and implicit methods for ordinary differentialjequations were discussed in §16.6.) The FTCS algorithm is also an example ofa single-level scheme, since only values at time level n have to be stored to findvalues at time level n + 1.von Neumann Stability AnalysisUnfortunately, equation (19.1.11) is of very limited usefulness.
It is an unstablemethod, which can be used only (if at all) to study waves for a short fraction of oneoscillation period. To find alternative methods with more general applicability, wemust introduce the von Neumann stability analysis.The von Neumann analysis is local: We imagine that the coefficients of thedifference equations are so slowly varying as to be considered constant in spaceand time. In that case, the independent solutions, or eigenmodes, of the differenceequations are all of the formunj = ξ n eikj∆x(19.1.12)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).x or j83719.1 Flux-Conservative Initial Value ProblemsLaxt or nFigure 19.1.2. Representation of the Lax differencing scheme, as in the previous figure. The stabilitycriterion for this scheme is the Courant condition.where k is a real spatial wave number (which can have any value) and ξ = ξ(k) isa complex number that depends on k.
The key fact is that the time dependence ofa single eigenmode is nothing more than successive integer powers of the complexnumber ξ. Therefore, the difference equations are unstable (have exponentiallygrowing modes) if |ξ(k)| > 1 for some k. The number ξ is called the amplificationfactor at a given wave number k.To find ξ(k), we simply substitute (19.1.12) back into (19.1.11). Dividingby ξ n , we getξ(k) = 1 − iv∆tsin k∆x∆x(19.1.13)whose modulus is > 1 for all k; so the FTCS scheme is unconditionally unstable.If the velocity v were a function of t and x, then we would write vjn in equation(19.1.11). In the von Neumann stability analysis we would still treat v as a constant,the idea being that for v slowly varying the analysis is local.
In fact, even in thecase of strictly constant v, the von Neumann analysis does not rigorously treat theend effects at j = 0 and j = N .More generally, if the equation’s right-hand side were nonlinear in u, then avon Neumann analysis would linearize by writing u = u0 + δu, expanding to linearorder in δu. Assuming that the u0 quantities already satisfy the difference equationexactly, the analysis would look for an unstable eigenmode of δu.Despite its lack of rigor, the von Neumann method generally gives validanswers and is much easier to apply than more careful methods.
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