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514 chapter 196. Now modify your program to solve Burgers’ shock equation (19.8) usingthe Lax–Wendroff method (19.12).7. Save the initial data and the solutions for a number of times in separate filesfor plotting.8. Plot the initial wave and the solution for several time values on the samegraph in order to see the formation of a shock wave (like Figure 19.1).9. Run the code for several increasingly large CFL numbers. Is the stabilitycondition β
solitons & computational fluid dynamics 515dispersion may arise from terms in the wave equation that contain higher-orderspace derivatives. For example, consider the waveformu(x, t)=e ±i(kx−ωt) (19.13)corresponding to a plane wave traveling to the right (“traveling” because thephase kx − ωt remains unchanged if you increase x with time). When this u(x, t) issubstituted into the advection equation (19.4), we obtainω = ck. (19.14)This equation is an example of a dispersion relation, that is, a relation betweenfrequency ω and wave vector k. Because the group velocity of a wavev g = ∂ω∂k , (19.15)the linear dispersion relation (19.14) leads to all frequencies having the same groupvelocity c and thus dispersionless propagation.Let us now imagine that a wave is propagating with a small amount of dispersion,that is, with a frequency that has somewhat less than a linear increase with the wavenumber k:ω ≃ ck − βk 3 . (19.16)Note that we skip the even powers in (19.16), so that the group velocity,v g = dωdk ≃ c − 3βk2 , (19.17)is the same for waves traveling to the left the or the right. If plane-wave solutionslike (19.13) were to arise from a wave equation, then (as verified by substitution)the ω term of the dispersion relation (19.16) would arise from a first-order timederivative, the ck term from a first-order space derivative, and the k 3 term from athird-order space derivative:∂u(x, t)∂t+ c∂u(x, t)∂x+ β ∂3 u(x, t)∂x 3 =0. (19.18)We leave it as an exercise to show that solutions to this equation do indeed havewaveforms that disperse in time.19.5 Shallow-Water Solitons, the KdeV EquationIn this section we look at shallow-water soliton waves. Though including some complications,this subject is fascinating and is one for which the computer has beenabsolutely essential for discovery and understanding. In addition, we recommendthat you look at some of the soliton animation we have placed in the animations folderon the CD.C D−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 515
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COPYRIGHT 2008, PRINCET O N UNIVE R
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−101COPYRIGHT 2008, PRINCET O N U
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Copyright © 2008 by Princeton Univ
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viiicontents2.3 Experimental Error
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xcontents6.3 Experimentation 1356.4
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xiicontents9.7.1 Friction: Model an
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xivcontents12.8 Signals of Chaos: L
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xvicontents14.15.2Exercise 2: Cache
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xviiicontents18.4 Waves for Variabl
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xxcontentsC.3 DX Tools Summary 576C
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xxivprefacewhich includes understan
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2 chapter 1PhysicsCPFigure 1.1 A re
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4 chapter 1Scientific LibrariesPerf
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6 chapter 1C DWe have tried to make
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8 chapter 1possibly when installing
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10 chapter 11.4.1 Structured Progra
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12 chapter 17. Revise Area.java so
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14 chapter 1argv). Because main met
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16 chapter 1Package Class Tree Depr
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18 chapter 1Memory and storage size
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20 chapter 1TABLE 1.4The IEEE 754 S
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22 chapter 1gives 24-bit precision
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24 chapter 1TABLE 1.6Representation
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26 chapter 1The computer fetches th
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28 chapter 1Your problem is to use
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2Errors & Uncertainties in Computat
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32 chapter 2purposes, let us consid
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34 chapter 2can be avoided by combi
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36 chapter 2means that a program ru
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38 chapter 2To be more specific, le
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40 chapter 2Let us assume that an a
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42 chapter 2To see if these assumpt
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44 chapter 2computation. Accordingl
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46 chapter 3is regular practice to
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48 chapter 3to plot for x and y, we
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50 chapter 3Sample PtPlot Data file
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52 chapter 3TABLE 3.1Text Files Gra
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54 chapter 3Figure 3.4 Left: A plot
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56 chapter 3TABLE 3.2Grace Menu and
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58 chapter 33.4.1 Gnuplot Input Dat
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60 chapter 3gnuplot> set output "pl
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62 chapter 3gnuplot> set terminal e
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64 chapter 3By setting terminal to
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66 chapter 33.6 Texturing and 3-D I
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68 chapter 4R R 2RL L 2LC C 2CFigur
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70 chapter 44.3 Resistance Becomes
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72 chapter 4be either static or dyn
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74 chapter 4On line 8 we see a meth
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76 chapter 44.4.3 Static and Nonsta
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78 chapter 42. Compile and execute
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80 chapter 41 / Z0.51400R0400R80021
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82 chapter 4no arguments and return
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84 chapter 42. Define a daughter cl
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86 chapter 4new features without
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88 chapter 4qFigure 4.7 Left: The t
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90 chapter 4with properties differi
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92 chapter 4In a project such as th
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94 chapter 4data types called objec
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96 chapter 46. Change the mass of t
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98 chapter 4Check that all the plot
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100 chapter 43. You should see now
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102 chapter 44.9.11 Complex Object
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104 chapter 4✞/ / KomplexTest : t
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106 chapter 4motion in other direct
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108 chapter 4✝protected double y(
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110 chapter 5Mathematically, the li
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112 chapter 5The linear congruent m
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114 chapter 5TABLE 5.1A Table of a
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116 chapter 55.3 Unit II. Monte Car
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118 chapter 5300y0-10-20R2001000 20
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120 chapter 54100,000log[N(t)]10,00
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122 chapter 5✞/ / Decay . java :
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124 chapter 6f(x)a x i x i+1 x i+2
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126 chapter 6f(x)f(x)parabola 1para
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128 chapter 6evaluate the function
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130 chapter 6⇒ N =( ) 2/91 1(ɛ m
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132 chapter 63. [−∞→∞], sca
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134 chapter 6}public static double
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136 chapter 6the integral of f(x)=1
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138 chapter 6f(x)< f(x) >xFigure 6.
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140 chapter 61. Conduct 16 trials a
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142 chapter 6acceptrejectFigure 6.6
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144 chapter 6The crux of this techn
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7Differentiation & SearchingIn this
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148 chapter 7the forward-difference
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150 chapter 7algorithm (7.7) is O(h
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152 chapter 7algorithms in which de
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154 chapter 7we know a zero occurs.
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156 chapter 7Figure 7.3 Two example
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8Solving Systems of Equationswith M
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160 chapter 8the spheres, and the d
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162 chapter 8We now have a solvable
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164 chapter 8of (8.19) by A −1 :x
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166 chapter 8Row MajorColumn Majora
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168 chapter 8having different varia
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170 chapter 8Matrix objects, add an
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172 chapter 8✞/∗ JamaFit : JAMA
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174 chapter 8( ) α β3. Consider t
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176 chapter 8discarding some inform
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178 chapter 8Lagrange interpolation
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180 chapter 8Figure 8.3 Three fits
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182 chapter 8if you have the abilit
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184 chapter 840fitNumber20dataN(t)0
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186 chapter 8theoretical curve went
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188 chapter 8This is a measure of t
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190 chapter 88.7.4 Linear Quadratic
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192 chapter 8Your problem here is t
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9Differential Equation Applications
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196 chapter 9V(x)HarmonicAnharmonic
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198 chapter 9B(t) are solutions of
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200 chapter 9This expresses the acc
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202 chapter 9derivative:dy(t)dt≃
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204 chapter 9As an example of the u
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206 chapter 9C Dhigh-precision work
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208 chapter 9Amplitude Dependence,
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210 chapter 9Here N is the normal f
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212 chapter 99.9 Unit II. Binding A
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214 chapter 9the Schrödinger equat
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216 chapter 99.11.1 Numerov Algorit
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218 chapter 9}}xl = xl0+i∗h;ul[i]
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220 chapter 9public static double d
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222 chapter 99.13 Unit III. Scatter
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224 chapter 9The equations for the
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226 chapter 9Figure 9.9 The traject
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228 chapter 9fFigure 9.10 Left: The
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230 chapter 99.17.1.1 EXPLORATION:
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232 chapter 10y(t)10Y( )10-1t-10 20
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236 chapter 1010.4 Fourier Transfor
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242 chapter 10102 4 6-1Figure 10.2
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246 chapter 104. As always, check t
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248 chapter 10A special case of the
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254 chapter 10A 1.5mp1.0lit 0.5ud 0
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256 chapter 1010.8 Unit III. Fast F
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258 chapter 10Figure 10.9 The basic
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260 chapter 10TABLE 10.1Binary-Reve
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262 chapter 10✞/∗ FFT. java : F
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11Wavelet Analysis & Data Compressi
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266 chapter 11the second derivative
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268 chapter 1111.2.1 Wave Packet As
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270 chapter 11localized in time, su
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272 chapter 11201 Y100Signal0-10-10
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278 chapter 11FrequencyTimeFigure 1
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280 chapter 11N SamplesInputN/2(1)c
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282 chapter 11Figure 11.10 The filt
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wavelet analysis & data compression
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288 chapter 113. Reproduce the scal
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290 chapter 12discrete decay law,
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292 chapter 120.80.400 10 20Ax nn n
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296 chapter 121b=1.0b=4.0b=5.0X0-1-
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300 chapter 12✞/ / LyapLog . java
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302 chapter 1212.10 Unit II. Pendul
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304 chapter 12In Chapter 9, “Diff
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306 chapter 12Figure 12.6 The data
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308 chapter 12rotating solutionsθ2
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310 chapter 12θ(t)8040.θ(0)=0.314
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312 chapter 12Figure 12.11 Top row:
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314 chapter 12|θ(t)|200 1 2Figure
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Angular Velocity of Lower Pendulum3
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318 chapter 12some fraction of a ch
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320 chapter 1212.19 Lotka-Volterra
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322 chapter 12numbers that accounts
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324 chapter 12400200pPopulationpP00
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13Fractals & Statistical GrowthIt i
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328 chapter 1330010,000 points20010
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330 chapter 13Figure 13.2 Left: A f
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332 chapter 13copy of a frond, and
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334 chapter 13The results of this t
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336 chapter 13log N(r) = log C −
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338 chapter 13to determine the slop
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340 chapter 13✞☎import java . i
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342 chapter 13Figure 13.8 Number 8
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344 chapter 13Conway in the 1970s,
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346 chapter 13(x 0, y 1)(x 0, y 0)(
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348 chapter 13yxFigure 13.13 After
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350 chapter 13gradient // Vertical
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14High-Performance Computing Hardwa
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354 chapter 14A(1)A(2)A(3)CPUA(N)M(
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356 chapter 14ADBCFigure 14.4 Multi
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358 chapter 14as Fortran or C, tran
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360 chapter 14TABLE 14.2Computation
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362 chapter 14The processors in a p
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364 chapter 14Figure 14.7 Two views
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366 chapter 14Speedup8Amdahl's Lawp
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368 chapter 1414.11 Parallelization
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370 chapter 14The problem affects p
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372 chapter 14• A race condition
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374 chapter 14yet in order to obtai
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376 chapter 14slightly faster or sm
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378 chapter 14You see (Listing 14.1
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380 chapter 14/optimize:4/optimize:
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382 chapter 14• As indicated in
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384 chapter 1410,000Execution Time
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386 chapter 14as high-performance c
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388 chapter 14✞Dimension Vec( idi
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16Simulating Matter withMolecular D
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426 chapter 16+ +Figure 16.1 The mo
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428 chapter 16the molecules stay cl
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430 chapter 162 3 2 3 2 31 4 1 4 1
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432 chapter 16Velocity-Verlet Algor
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434 chapter 16Energy vs Timefor 36
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436 chapter 16Figure 16.7 A simulat
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thermodynamic simulations & feynman
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17PDEs for Electrostatics & Heat Fl
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pdes for electrostatics & heat flow
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Page 549 and 550: solitons & computational fluid dyna
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Page 595 and 596: Appendix A: Glossaryabsolute value
Page 597 and 598: glossary 557control character — A
Page 599 and 600: glossary 559log in (on) — To sign
Page 601 and 602: glossary 561supercomputer — The c
Page 603 and 604: installing ptplot & java developer
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installing ptplot & java developer
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installing ptplot & java developer
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industrial-strength data visualizat
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Appendix D: An MPI TutorialIn this
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an mpi tutorial 595• Open a shell
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an mpi tutorial 597of all the compu
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an mpi tutorial 599TABLE D.1Some Co
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an mpi tutorial 601jobs to MPI. 2 A
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an mpi tutorial 603✞> cd $HOME Do
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an mpi tutorial 605✞/ / MPIhello
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an mpi tutorial 607Argument Namemsg
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an mpi tutorial 609D.3.4 Broadcast
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an mpi tutorial 611D.4 Parallel Tun
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an mpi tutorial 613✝}MPI_Recv ( &
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an mpi tutorial 615✞/ / Code list
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an mpi tutorial 617-1) does not sen
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an mpi tutorial 619D.6.1 Nonblockin
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an mpi tutorial 621D.9 List of MPI
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Appendix E: Calling LAPACK from CCa
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calling LAPACK from C 625E.2 Compil
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software on the CD 627JavaCodes Con
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software on the CD 629JavaCodes Con
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software on the CD 631Applets Direc
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software on the CD 633Fortran77code
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Appendix G: Compression via DWTwith
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compression via DWT with thresholdi
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compression via DWT with thresholdi
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BIBLIOGRAPHY[Abar 93] Abarbanel, H.
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ibliography 643[Erco] Ercolessi, F.
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ibliography 645[L&F 93] Landau, R.
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ibliography 647[P&W 91] Pinson, L.
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ibliography 649[VdeV 94] van de Vel
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IndexAbstract data structures,70Abs
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index 653drand, 114Driving force, 2
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index 655Libraries, see Subroutines
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index 657structured, 10for virtual
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