Ph 20.3 Numerical Solution of Ordinary Differential Equations

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1 Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 -v This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing Lab, and to introduce you to the basic process of coputational physics. In particular you should now be failiar with the basic idea: taking soe physical or atheatical syste forulated in ters of equations, ipleenting it using Python code, and plotting the resulting output. The purpose of this assignent is to deepen your understanding of how to investigate a syste once you ve ipleented it in code. In the first assignent, you already saw that it was soeties appropriate to think of a Lissajous figure as a function of tie and soeties ore useful to think of it as tracing out a fixed curve in the plane. In this assignent we will explore one of the ost iportant odel systes in all of physics: the siple haronic oscillator. In order to do this we ll ipleent several ethods for the nuerical solution of siple differential equations. If you have already studied differential equations in a course on calculus, you ay want to review the for this (and future) assignents; if you haven t yet, don t worry: differential equations are just a fancy nae for equations that contain derivatives. Just like in the last assignent, because we know the analytic behavior of the syste we can evaluate the behavior of the three different ethods we ll use. While soe aspects of this behavior will be obvious just fro plotting the various solutions directly (i.e., fro plots of position and velocity as functions of tie), we can gain a better understanding by looking at the energy as a function of tie or by looking at the geoetry of the phase-space curve of position versus velocity traced out as a function of tie. The basic coding task of this assignent is to ipleent equations (7), (8), and (12). These three equations only differ by what s written in their subscripts, but you ll see they all have very different behavior, so it s very iportant to ake sure you understand how to translate the to code in order to ipleent the correctly. In the previous two assignents, we were able to avoid loops entirely because each point in the array could be acted on independently, and therefore at the sae tie using nupy routines. In this assignent this is no longer the case: e.g. the point x i+1 depends explicitly on the previous point x i. So you will need to use loops. You are of course free to continue using nupy arrays rather than Python lists! Note that it is in general uch faster for a coputer to alter the values of an array of fixed size rather than to append new values to the end of an array (talk to your TA if you d like to know ore!). So if, like in this assignent, you know how long your list or array will end up being you ay wish to create a duy list/array of that size to start with using e.g. [0]*n or np.zeros(n) and then alter its values as you go. After copleting the first three assignents of Ph 20 you should feel very cofortable using Python to create, plot, and analyze nuerical data. Starting with the next assignent we will step away fro Python to introduce other useful scientific coputing tools: akefiles, version control, and Matheatica. We will return briefly to Python in the last assignent of the ter. The reaining two ters in the Ph 2x sequence will use Python exclusively: Ph 21 for data analysis, Ph 22 for nuerical ethods. 1

2 An ideal, horizontal spring with a ass attached to its free end oscillates freely, displacing the ass a variable distance, x, with tie. The equilibriu position is that for which the spring is neither copressed nor extended. The force on the ass is proportional to the displaceent x of the ass,, fro its equilibriu position: x x F = a = kx. Figure 1: The ipleentation of a siple haronic oscillator as a ass on a spring. Ordinary Differential Equations in Mechanics Newton s second law is force = ass acceleration, or (for the otion of a syste with one degree of freedo) [ F = a = dv ] dt = x d2 dt 2, (1) where x, v, a, and t are respectively the position, velocity, accelation, and of course tie. If the acceleration, a, or the velocity, v, are known explicitly as functions of tie alone, we can obtain the position, x, very siply. [One would say that the proble is reduced to quadratures, which is a fancy way of saying that the solution can be written explicitly as an integral of known functions.] We get out our Sipson routine, and integrate. If the velocity is known explicitly, if the acceleration is known explicitly, first integrate t x(t) = x 0 + v(t ) dt ; (2) 0 t v(t) = v 0 + a(t ) dt (3) 0 to get the velocity at a set of equally spaced ties; then integrate again to get the position. In ost interesting probles, however, acceleration and velocity are not known beforehand as functions of tie alone: rather, the expression for the force ay involve the position and/or velocity theselves (we have already seen this in the grapefruit proble in assignent 1). Consequently, the straightforward integration of the acceleration or velocity is not a viable route. Enter nuerical techniques for the integration of ordinary differential equations. We shall try out the siplest of these on a a very siple syste: the siple haronic oscillator, as exeplified by the case of a ass attached to a spring (Fig. 1). 2

3 Errors in nuerical solutions It is iportant to be aware of two types of errors that arise when we use coputers to solve atheatical probles. The first is roundoff error, which occurs because coputers use finite representations of real nubers 1. For instance, the real nuber 2 has a decial representation with an infinite nuber of digits; however, a coputer has to ake do with (typically) 64 bits of inforation to store this nuber, corresponding to (roughly) 15 decial digits. One exaple of roundoff error is coputing : the coputer sees only the first 15 or so digits of the nuber 3, so the answer to will siply be 3. (There do exist arbitrary precision nuerical routines that can store nubers to any nuber of digits required for a given accuracy; for exaple, section 20.6 of Nuerical Recipes or soe algoriths in Matheatica. However, these routines are typically coplicated and are uch uch slower than finite representation arithetic). Another kind of coputational error is truncation error, which is introduced when we replace procedures such as differentiation and integration, which are based on liits, with finite representations of these procedures. This all sounds very abstract, so here is an exaple. The derivative f (x) of a real function is defined by df(x) dx = li h 0 f(x + h) f(x), (4) h and the liit (a sybolic operation) tells us what happens to the ratio on the right hand side of the equation when h 0. On a coputer, we can do soething like df(x) dx f(x + h) f(x), (5) h and we choose h sall, but still finite (so that you don t divide by zero!). The sybol is there to point out that the equation is only approxiate, because the exact definition of the derivative requires that you really take the liit as h 0. The error you ake here is called truncation error, which is different than roundoff error. Truncation error depends not on properties of the coputer eory, but on the value that you choose for h. The nae truncation error coes fro the ost usual source of such error, which is truncating an infinite Taylor series at a finite nuber of ters. We have already seen truncation error in action in the last assignent where we copared Sipson s and Trapezoidal ethods of coputing definite integrals. Note that if h is too sall, Eq. (5) will not work because of roundoff error (think about what happens if x is 3 and h is ). One of the hardest jobs in coputational science is to choose paraeters like h sall enough that your solutions are acceptable (that is, truncation error is sall enough), but large enough that your coputations are still viable on the coputer you have at hand (that is, roundoff error is sall enough). Integration of ordinary differential equations Turning Eq. (5) around, we get a siple prescription to integrate nuerically the first-order differential equation df/dx = g(x): f(x + h) f(x) + hg(x). (6) 1 We are here ignoring sybolic coputation (e.g. Matheatica), and we are considering instead nuerical coputation in a language such as Python or C++ 3

4 If you are given f 0 f(x 0 ) at the initial x = x 0, you can apply (6) repeatedly to obtain the values f i f(x 0 + ih), for as any steps i as you wish. Copare what we have just done with its analytic analogue: analytic integration (when it is possible) yields the sybolic representation of f(x), given the initial value f(x 0 ) and the sybolic representation of g(x); nuerical integration yields the approxiate values f i f(x i ) at the discrete points x i, given f(x 0 ) and g(x) [we are happy to work with a sybolic representation of g(x), but we what we really need are only the values g i g(x i )]. Euler ethods for the spring Going back to the spring, a let us apply the ethod that we have just learned. For siplicity, we shall set k/ = 1 (where k is the spring constant and is the ass); if you re bothered that this isn t general you can convince yourself that we can always do this by an appropriate choice of units. We want to obtain the approxiate values x i x(t i ) and v i v(t i ) for t i = t 0 + ih, with i = 1,..., N [careful with the notation, here x becoes a dependent variable, while t is the new independent variable]. We get to pick the initial conditions x 0 and v 0, and for siplicity we also set t 0 = 0. The straightforward application of Eq. (6) then yields the explicit Euler ethod, x i+1 = x i + hv i, v i+1 = v i hx i. (7) Here the old position and velocity are used to copute, respectively, the new velocity and position. It should be clear that this choice is arbitrary. In fact, we could also have used an iplicit Euler ethod, x i+1 = x i + hv i+1, v i+1 = v i hx i+1, (8) where the new position and velocity are used to copute the new velocity and position. In this case, we need the unknown values x i+1 and v i+1 to update the known values x i and v i, so Eq. (8) encodes the linear syste ( ) ( ) ( ) 1 h xi+1 xi =, (9) h 1 v i+1 which we need to solve before we can get x i+1 and v i+1 as functions of x i and v i. The solution is easy to obtain in the case of the spring (and you will work it out in this week s assignent), but for any differential equations the discrete syste (9) is not linear, which akes its solution difficult. This is why in practice explicit ethods [generalizations of (7)] are used ore often than iplicit ethods [generalizations of (8)]. Assignent (Part 1) 1. Write a progra in Python to investigate nuerically the otion of a ass on a spring, ipleenting the explicit Euler ethod. Plot x and v as functions of tie for a few cycles of oscillation, choosing h sall enough that the graph looks sooth, but not saller. Choose reasonable initial conditions. 2. Work out the analytic solution to this proble for your initial conditions, and copare it to your nuerical solution by plotting the global errors x analytic (t i ) x i and v analytic (t i ) v i. Choose h so that the errors becoe apparent within a few cycles of oscillation. 3. Show that the truncation error is proportional to h for reasonably sall values of h. One way to do this is to plot the axiu value of x analytic (t i ) x i versus h for several runs v i 4

5 of different h integrating up to the sae final tie. You need only a few values of h, say h = h 0, h 0 /2, h 0 /4, h 0 /8 and h 0 / Now copute the nuerical evolution of the noralized total energy E = x 2 + v 2 (which should be conserved in this physical syste), and plot E as a function of tie. What is the long-range trend for E? How does it copare with the evolution of the global errors? 5. Solve the syste (9) to obtain equations siilar to (7) for the iplicit Euler ethod (you need expressions for x i+1 and v i+1 in ters of x i and v i only). Ipleent the iplicit Euler ethod nuerically, and investigate how the global errors and the evolution of the energy change with respect to the explicit Euler ethod (to draw a fair coparison, use the sae h for both ethods). 6. When you are done with all this, read on. The conservation of energy and otion in phase space Fair enough: we had been cautioned that Eq. (6) was only approxiate, and that it would introduce errors in our solutions. Still, it is troubling to find that the cherished principle of energy conservation does not hold for the nuerical spring. A odern way to look at this kind of probles is the following. When we write the finite-difference equations (7) and (8), we do soething very drastic: we take a differential equation that has any properties (including the fact that its analytic solutions conserve energy) and we replace it with a very different beast, a discrete iterative ap. This is essentially a rule (call it F ) to produce the values x i+1 and v i+1 fro the values x i and v i, as in (x i+1, v i+1 ) = F [(x i, v i )]; (10) or, equivalently, to produce the n th values x n and v n fro the n th iteration of the ap, (x n, v n ) = F (n) [(x 0, v 0 )] = F [F [... F [(x 0, v 0 )]...]]. (11) }{{} n ties Depending on how we generate F fro the particular differential equation that we are trying to solve, the iterative ap can have very different properties fro the original analytic equation. For instance, in the case of the spring, the explicit and iplicit Euler ethods generate iterative aps that do not conserve energy (that is, they are dissipative). There is a branch of nuerical analysis (known as backward error analysis) that studies the behavior of nuerical ethods for differential equations by building the iterative aps, and then finding the analytic differential equations (typically odifications of the original ones) that the iterative aps are actually solving exactly. Because error has to creep in soewhere (we never said coputers are perfect!), you will seldo be able to find a ap that yields exact solutions for the very equation you want; but you can try to arrange things so that the odified equation that you are really solving is not so different, at least with regard to its crucial properties. So what are the crucial properties that physicists are ostly concerned with? This is a hard question! We can give a general answer for a very broad class of physical systes of interest to physicists, those described by Hailton s equations. You will eet the Hailtonian foralis later in the course of your studies; suffice it to say now that the dynaics of these systes can be derived fro a variational principle, which states that the physical solutions are found to be the functions that extreize certain physical quantities, such as the the tie integral of the total energy. The siple haronic oscillator is one such syste. 5

6 It is especially instructive to study the behavior of Hailtonian systes in phase space, whose diensions are (roughly speaking) all the generalized 2 positions of the syste and all the corresponding generalized velocities. For the one-diensional spring, phase space is just the (x, v) plane. Any iaginable evolution of the spring appears as a trajectory (x(t), v(t)) in phase space, but the actual physical solutions are all circles, defined by x 2 + v 2 = E; that is, the physical trajectories trace out the curves of constant energy! Syplecticity (that is, conservation of volue in phase space) is a characteristic property of Hailtonian systes. What does it ean to conserve volue (or, for the spring, area) in phase space? Suppose we are given a group of springs with slightly different initial conditions, so that at the tie t 0 they occupy the region S 0 in phase space alost uniforly; at the later tie t 1, the springs will have all evolved to occupy a different region S 1 in phase space, but the area of S 1 will be the sae as the original area of S 0. In the nuerical arena, one can build syplectic nuerical integrators that, indeed, conserve area in phase space. In doing so, they produce solutions that, at least qualitatively, look ore physical than those produced by nonsyplectic integrators (for instance, on the average they conserve energy). A very siple syplectic integrator is a cross (of sorts) between the explicit and iplicit Euler ethods: x i+1 = x i + hv i, v i+1 = v i hx i+1 ; (12) this is the syplectic Euler ethod. Backward error analysis would show that (12) solves exactly a odified analytic syste that is (first of all) Hailtonian, and (second) uch closer to original syste. This is the reason why syplectic integrators are often used for Hailtonian systes. They are not, however, the solution to all evils, because of what I call the conservation of pain or that is soeties also called the blanket principle (the blanket is always too short to cover both your feet and your head). Error will spare the syplectic structure of phase space, but it ust reappear elsewhere: for instance, syplectic integrators ight preserve better the aplitude of the spring oscillations, which appears in the expression for the energy; but the precise position of the ass within each oscillation (i.e., its phase) will accuulate a growing error with respect to the exact solution. Assignent (Part 2) 1. Investigate the phase-space geoetry of the trajectories produced by the explicit and iplicit Euler ethods. Set h so that deviations fro closed circles (i.e. the analytic solutions, which you should also plot) are obvious. 2. Ipleent the syplectic Euler ethod in Python and investigate the phase-space geoetry of the trajectories that it produces. Copare the with the trajectories obtained at the previous point, for the sae h. This is your Ph20 Beautiful Plot TM of the week. 3. Study the evolution of total energy obtained with the syplectic Euler ethod. Set h so that deviations fro the constant value of the analytic solution are obvious. What do they look like? How does this evolution relate to what you just saw in phase space? 4. (optional) Exaine the long-ter evolution of the global error in the phase for trajectories produced by the syplectic Euler ethod, as copared to the exact solution. To do so, plot 2 In the Hailtonian foralis it is often useful to use position-like variables that do not correspond directly to easureents of distance. 6

7 the syplectic-euler solution on top of the exact solution, and see if the oscillations develop a lag. Many oscillations ay be needed for the error to grow appreciably; plot only the last few (of any) to estiate the lag visually. If this assignent was interesting to you, consider taking Ph 22, which takes the knowledge obtained fro this assignent as a starting point and builds up to full N-body siulations of gravitationally-interacting objects. 7

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