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Chapter 5. Dissipation, Dispersion, and Group Velocity

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Chapter 5. Dissipation, Dispersion, and Group Velocity CHAPTER TREFETHEN Chapter Dissipation Disp ersion and Group Velo city Disp ersion relations Dissipation Disp ersion and group velo city Mo died equations p Stabilityin norms Notes and references Things fall apart the center cannot hold Mere anarchy is loosed upon the world WBYEATS The Second Coming CHAPTER TREFETHEN It would be a ne thing if discrete mo dels calculated solutions to partial dierential equations exactly but of course they do not In fact in general they could not even in principle since the solution dep ends on an innite amount of initial data Instead the b est we can hop e for is that the errors intro duced by discretization will be small when those initial data are reasonably well b ehaved This chapter is devoted to understanding the b ehavior of numerical errors From truncation analysis wemayhave a b ound on the magnitude of discretiza tion errors dep ending on the step sizes h and k but much more can be said for the b ehavior of discretization errors exhibits great regularitywhichcanbe quantied by the notions of numerical dissipation and disp ersion Rounding errors to o though intro duced essentially at random propagate in the same predictable ways So long as we can estimate the magnitude of the discretization and round ing errors what is the p oint in trying to investigate their behavior in more detail There are several answers to this question One is that it is a go o d idea to train the eye a practitioner familiar with articial numerical eects is less likely to mistake spurious features of a numerical solution for mathematical or physical reality Another is that in certain situations it may be advantageous to design schemes with sp ecial prop ertieslow dissipation for example or low disp ersion A third is that in more complicated circumstances the mag nitude of global errors may dep end on the behavior of lo cal errors in ways that ordinary analysis of discretization and rounding errors cannot predict In particular we shall see in the next chapter that the stability of b oundary con ditions for hyp erb olic partial dierential equations dep ends up on phenomena of numerical disp ersion One mightsay that this chapter is built around an irony nite dierence approximations have a more complicated physics than the equations they are designed to simulate The irony is no paradox however for nite dierences are used not b ecause the numbers they generate have simple prop erties but b ecause those numb ers are simple to compute DISPERSION RELATIONS TREFETHEN Disp ersion relations Any timedep endent scalar linear partial dierential equation with con stant co ecients on an unb ounded space domain admits plane wave solutions ixt ux te R where is the wave number and is the frequency Vector dierential equations admit similar mo des multiplied by a constant vector the extension to multiple space dimensions is describ ed at the end of this section For each not all values of can be taken in Instead the PDE imp oses a relationship b etween and which is known as the disp ersion relation mentioned already in x In general each wave number corresp onds to m frequencies where m is the order of the dierential equation with resp ect to t and that is why is called a relation rather than a function For most purp oses it is appropriate to restrict attention to values of that are real in whichcase mayberealor complex dep ending on the PDE The wave decays as t if Im and grows if Im For example here again are the disp ersion relations for the mo del equa tions of x and also for the secondorder wave equation u u t x u u ie tt xx u u i t xx u iu t xx These relations are plotted in Figure Notice the doublevaluedness of the disp ersion relation for u u and the dashed curve indicating complex tt xx values for u u t xx More general solutions to these partial dierential equations can be ob tained by sup erimp osing plane waves so long as each comp onent sat ises the disp ersion relation the mathematics behind such Fourier synthesis DISPERSION RELATIONS TREFETHEN a u u b u u t x tt xx i c u u d u iu t xx t xx Figure Disp ersion relations for four mo del partial dierential equations The dashed curve in c is a reminder that is complex DISPERSION RELATIONS TREFETHEN was describ ed in Chapter and examples were given in x For a PDE of rst order in t the result is Z ix t ux t e u d Since most partial dierential equations of practical imp ortance have variable co ecients nonlinearity or b oundary conditions it is rare that this integral representation is exactly applicable but it may still provide insightinto lo cal b ehavior Discrete approximations to dierential equations also admit plane wave solutions at least if the grid is uniform and so they to o have disp ersion relations To b egin with let us discretize in x only so as to obtain a semidis crete formula Here are the disp ersion relations for the standard centered semidiscretizations of u u sin h t h h u u sin tt h h u u i sin t h h u i u sin t h These formulas are obtained by substituting into the nite dierence formulas with x x In keeping with the results of x each disp ersion j relation is hp erio dic in and it is natural to take h h as a fundamental domain The disp ersion relations are plotted in Figure sup erimp osed up on dotted curves from Figure for comparison Stop for a moment to compare the continuous and semidiscrete curves in Figure In each case the semidiscrete disp ersion relation is an accurate approximation when is smal l which corresp onds to many grid points per wavelength The numberofpoints p er spatial wavelength for the wave is h In general the disp ersion relation for a partial dierential equation is a p olynomial relation between and while a discrete mo del amounts to a trigonometric approximation Although other design principles are p ossible the standard discrete approximations are chosen so that the trigonometric function matches the p olynomial to as high a degree as p ossible at the origin To illustrate this idea Figure plots disp ersion relations for the standard semidiscrete nite dierence approximations to u u and u iu t x t xx of orders and The formulas were given in x DISPERSION RELATIONS TREFETHEN a u u b u u t tt i c u u d u i u t t Figure Disp ersion relations for centered semidiscrete approx imations to the four mo del partial dierential equations Each func tion is hp erio dic in the plots show the fundamental domain h h DISPERSION RELATIONS TREFETHEN a u u b u iu t x t xx Figure Disp ersion relations for semidiscrete centered dier ence approximations to u u and u iu of orders t x t xx Now let us turn to fully discrete nite dierence formulas discrete in time as well as space The p ossibilities b ecome numerous For example substituting the plane wave ix t n ij hnk n j v e e j into the leap frog approximation of u u yields the disp ersion relation t x i k i k i x i x e e e e where khthat is sin k sin h Similarly the CrankNicolson approximation of u u has the disp ersion t xx relation i k h i e i h i h i k e e e which reduces to k h i tan sin DISPERSION RELATIONS TREFETHEN These disp ersion relations are h p erio dic in and k p erio dic in With the use of inverse trigonometric functions it is p ossible to solve such equations for so as to exhibit the functional dep endence explicitlybut the resulting formulas are less app ealing and often harder to work with Equations like and have a certain eleganceone sees at a glance that the time and space derivatives have been replaced by trigonometric analogs Tables and consider once again the various nite dierence approximations to u u and u u that app eared in Tables t x t xx and In each case the disp ersion relation is b oth listed and plotted Since h and k are indep endent parameters there is now a range of p ossible plots wehave arbitrarily taken kh in the rst table and kh in the second That is whyeach plot o ccupies a rectangle of asp ect ratio in wavenumb er spacey Notice that the multistep leap frog formulas contain two branches of values For partial dierential equations in several space dimensions the notion of disp ersion relation generalizes just as x would suggest a plane wave takes the form i xt ux te R where and x are vectors and b ecomes a scalar function or relation of a vector argument For example the wave equation u u u tt xx yy has the disp ersion relation if the vector is written so that the lines of constant in the plane are concentric circles On the other hand the leap frog approximation n n n n n n n n v v v v v v v v ij ij ij ij ij ij ij ij appropriate to a uniform grid with h x y has the disp ersion relation i h h h k sin sin sin which is plotted in Figure for Once again the disp ersion relation is accurate for small wave numb ers but diverges dramatically elsewhere EXERCISES What are the co ecients as trigonometric functions of the disp ersion relations plotted in Figure Sketch the disp ersion relation for the leap frog mo del of u u with say t x How is the instability of this nite dierence formula reected in your sketch Not yet written y analogous to a Brillouin zone in crystallography C Kittel Intro duction to Solid State Physics Wiley DISSIPATION TREFETHEN i k BE Backward Euler i e sin h x k sin h CN CrankNicolson tan x
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