u t + f(u) x = 0, (12.1) f(u) x dx = 0. u(x, t)dx = f(u(a)) f(u(b)).

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1 12 Fiite Volume Methos Whe solvig a PDE umerically, how o we eal with iscotiuous iitial ata? The Fiite Volume metho has particular stregth i this area. It is commoly use for hyperbolic PDEs whose solutios ca spotaeously evelop iscotiuities as they evolve i time. These solutios are ofte calle shock waves. Coservatio Laws Cosier the coservatio law u t + f(u) x =, (12.1) where u is a (spatially) oe-imesioal coserve quatity, a f(u) is the flux of u. The cotiuous itegral formulatio of (12.1) states that b t a t b a u(x, t)x + b a f(u) x x =. u(x, t)x may be thought of as the time evolutio of the total mass of u across the omai [a, b], a is epeet oly o the flux through the bouaries, sice t b a u(x, t)x = f(u(a)) f(u(b)). This fact is a importat iea utilize by fiite volume methos, which geerally cosier the evolutio of u ot at a give poit, but istea i volume-average regios. For example, let {x i } be a gri of equally space poits with spacig x, a let C i be the i-th volume (subiterval) efie by (x i 1/2, x i+1/2 ). We are itereste i the evolutio of the volume average of u over this iterval, U i = 1 x C i u(x, t )x, where {t } is the time iscretizatio. The evolutio of these volume-average quatities will epe oly o the flux through the cell eges, so that u(x, t)x = f(u(x i 1/2, t)) f(u(x i+1/2, t)). (12.2) t C i 1

2 2 Lab 12. Fiite Volume Methos U i +1 t +1 F i 1/2 F i+1/2 t U i 1 U i U i+1 Figure 12.1: A schematic of the fluxes for the fiite volume metho as iicate by (12.3). We ca the costruct a time-steppig metho where i U i x (the total mass of the system) is coserve from oe time step to the ext. Let F i 1/2 = 1 t t +1 t t +1 t f(u(x i 1/2, t)) t. The [ ] u(x, t) x = u(x, t +1 ) u(x, t ) x, t C i C i ( ) = t Fi 1/2 F i+1/2. Thus, by itegratig (12.2) i time, we may approximate the evolutio of the cell ( volume ) averages with the metho where Ui = 1 x C i u(x, t ) t. U +1 i = Ui t ( ) Fi+1/2 x F i 1/2. (12.3) This formulatio guaratees the coservatio properties that are so esirable for coservatio laws, if the time-average fluxes Fi 1/2 ca be iscretize i a atural way. The key cotributio of fiite volume methos is the computatio of Fi 1/2. For a truly oliear f(u) this ca be rather complicate a messy, a typically will ivolve solvig what is usually referre to as the Riema problem for the coservatio law. The itereste stuet ca look at [Le22] for a very thorough itrouctio a iscussio o the subject. We will cosier the liear problem i oe imesio. The aalog to higher imesios is obtaie by cosierig the eigevector ecompositio of ay liear system. Noliear equatios complicate thigs further. The liear avectio equatio a upwiig The simplest coservatio law escribes the avectio or trasport of a quatity. The PDE is give by u t + au x =, (12.4) a escribes the motio of a cocetratio of some costituet u by a costat velocity oeimesioal wi a >.

3 3 I higher imesios this is a importat problem i may fiels, for example the trasport of chemicals i the atmosphere a oceas, proper mixig of various properties i metallurgy, a the passig of iformatio alog a etwork. Note that wheever u(x, t) is a solutio of the avectio equatio, the u(x at, t ) (for ay fixe t ) is also a solutio. Thus, if u(x, ) = u (x) the the solutio for all time ca be represete by u(x, t) = u (x at). This is a importat property of (12.4), a gives a ew meaig to the term avectio: this equatio merely takes the iitial coitios a passively trasports them with velocity a. For this equatio the computatio of the flux appears straightforwar: F i 1/2 = au i 1/2 where the U i 1/2 refers to the time average of U i 1/2 over the iterval t to t +1. Let us etermie how to approximate this time average. Note from Figure 12.2 that whe a > the flux that etermies U +1 i will be epeet o the value of Ui 1. Thus, oe possibility is to approximate the flux by F i 1/2 = au i 1. Usig this approximatio of the flux together with the flux ifferecig formula (12.3) yiels the first orer upwi metho, give by U +1 i = Ui a t ( U x i Ui 1 ). Aother way to erive the upwi metho is to istea suppose that what we wat to o is recostruct u(x) at each time step isie each cell (x i 1/2, x i+1/2 )from the mea values i that cell a its surrouig eighbors. This recostructe ũ(x) is the efie piecewise for each cell i. The solutio at the ext time step ca be fou as ũ +1 (x) = ũ (x a t) which allows us to etermie the fluxes F i 1/2 oce we have settle o a metho for etermiig ũ (x) i each cell. The simplest approach is ũ (x) = U i for x (x i 1/2, x i+1/2 ) This leas to fluxes give by F i 1/2 = a t = a t t+1 t t = au i 1. The followig coe solves the problem ũ (x i 1/2, t) t, (12.5) ũ (x i 1/2 at) t, (12.6) u t + au x =, < x < 1, u(x, t) = f(x), u(, t) = u(1, t), (12.7) where a ( ) (x.3) 2 f(x) = exp + χ.5 (.6,.7) x.6 χ (.6,.7.) = 1.6 < x <.7 x.7 Notice that this PDE has perioic bouary coitios. Essetially we are evolvig the sigal arou the uit circle. This allows us to evolve the sigal much further to test our umerical methos,

4 4 Lab 12. Fiite Volume Methos sice we oly have to iscretize the iterval [, 1] istea of a much larger omai. To see how to implemet the bouary coitios, cosier a gri = x < x 1 <... < x N 1 < x N = 1 of evely space poits. Sice u(x) is perioic the u(x N ) = u(x ), so it is sufficiet to track x,..., x N 1. import umpy as p from matplotlib import pyplot as plt from math import floor ef upwi(u, a, xmi, xmax, t_fial, t): """ Solve the avectio equatio with perioic bouary coitios o the iterval [xmi, xmax] usig the upwi fiite volume scheme. Use u as the iitial coitios. a is the costat from the PDE. Use the size of u as the umber of oes i the spatial imesio. Let t be the umber of spaces i the time imesio (this is the same as the umber of steps if you o ot iclue the iitial state). Plot a show the compute solutio alog with the exact solutio. """ t = float(t_fial) / t # Sice we are oig perioic bouary coitios, # we ee to ivie by u.size istea of (u.size - 1). x = float(xmax - xmi) / u.size lamba_ = a * t / x u = u.copy() for j i rage(t): # The Upwi metho. The p.roll fuctio helps us # accout for the perioic bouary coitios. u -= lamba_ * (u - p.roll(u, 1)) # Get the x values for the plots. x = p.lispace(xmi, xmax, u.size+1)[:-1] # Plot the compute solutio. plt.plot(x, u, label='upwi Metho') # Fi the exact solutio a plot it. istace = a * t_fial roll = it((istace - floor(istace)) * u.size) plt.plot(x, p.roll(u, roll), label='exact solutio') # Show the plot with the lege. plt.lege(loc='best') plt.show() # Defie the iitial coitios. # Leave off the last poit sice we're usig perioic # bouary coitios. x = 3 t = x * 3 // 2 x = p.lispace(., 1., x+1)[:-1]

5 5 u = p.exp(-(x -.3)**2 /.5) arr = (.6 < x) & (x <.7 ) u[arr] += 1. # Ru the simulatio. upwi(u, 1.2,, 1, 1.2, t) Try ruig the previous coe block with x set to 3, 6, 12, a 24. You will otice that the umerical solutio iffuses with time. It iffuses especially fast at the poits of iscotiuity. Piecewise liear recostructio a slope limiters The upwi metho is formally oly first orer, a actually oes relatively poorly i terms of actually trasportig the iitial ata with velocity a. You ca otice from the example coe that the upwi metho has errors that are iffusive meaig that the iitial ata is iffuse as time evolves, losig the peaks a fie etails. This is because the error for the upwi metho is o the orer of the seco erivative of u which is of a iffusive ature. To get a improve metho, cosier a better recostructio isie each cell, i.e. ũ (x) = U i + m i (x x i ) for x (x i 1/2, x i+1/2 ) (12.8) where the slope of this liear recostructio m i is etermie as a fuctio of the eighborig cell averages at time a Ui itself. The the flux is give by F i 1/2 = a t = a t ( = a t+1 t t ũ (x i 1/2 at) t, Ui 1 + m i (x i 1/2 at x i ), ) Ui 1 + m i 1 ( x a t). 2 (12.9) Oe of the most atural approaches is to just estimate the slope epeig o the cell i a a eighborig cell i + 1 or i 1. This leas to two popular methos, the Lax-Weroff metho a the Beam-Warmig metho (that really is the ame). The Lax-Weroff metho has a slope chose as m i = U i+1 U i x. (12.1) which it turs out is formally seco-orer accurate. It turs out though that the errors for this metho are ispersive, meaig that ear very steep graiets, the metho will geerate very rapi oscillatios (ue to the thir erivative of u ot beig approximate accurately). Aother way to cosier how these errors arise is to otice from Figure 12.2 that if the piecewise liear recostructio is avocate by some positive wi a the there will be places where the iscotiuous ature of the recostructio will itrouce spurious maxima or miima ito the solutio. These become the spurious waves see i simulatios usig the Lax-Weroff metho. A solutio to this ilemma betwee balacig the iffusive a ispersive errors comes from costructig slopes m i that esure o such o-mootoic trasport takes place. The basic iea is to costrai the slope so that the recostructe piecewise liear fuctio ũ (x) will ot geerate

6 6 Lab 12. Fiite Volume Methos U i+1 U i+2 U i U i 2 U i 1 Figure 12.2: The piecewise liear recostructio for the upwi a Lax-Weroff methos. The soli lies represet the simplest recostructio of the cell averages leaig to the upwi metho, a the ashe lies are those whose slope is obtaie via the Lax-Weroff metho. Note that the LW metho itrouces a spurious maximum at i + 3/2 (the cell ege betwee U i+1 a U i+2 ) a the miimum at i 3/2 will be uphysical exaggerate. The upwi metho avois this ifficulties, but clearly loses a sigificat amout of the available iformatio. This provies the motivatio for the slope limiters. uphysical extremal values whe it is avocate by some fiite wi a. The Mimo limiter chooses the slope as ( U m i = mimo i Ui 1, U i+1 U ) i (12.11) x x where a if a < b a ab > mimo(a, b) = b if b < a a ab > if ab <. (12.12)

7 Aalytic solutio Upwi Lax Weroff Mi mo Figure 12.3: Solutios of (12.7) at time t = 1.2 usig various methos. Here the avectio coefficiet is a = 1.2, a there are N = 1 subitervals i space, 15 subitervals i time. Problem 1. Implemet the Lax Weroff metho a use it to solve (12.7). For N = 3, 6, 12, 24, plot the aalytic solutio a the Lax-Weroff solutio. (You shoul have 4 separate plots, each with 3 graphs.) You shoul be able to tell that the Lax Weroff metho approximates the smooth portio of the sigal much better, as it oes ot struggle with iffusio. Ufortuately, it has some ifficulty with the iscotiuous portio, where uphysical oscillatios are see. Recall that we saw somethig similar i the waves lab whe there were iscotiuous iitial coitios. Hit: Use equatios 12.9 a Problem 2. Implemet the Mimo metho a use it to solve (12.7). For N = 3, 6, 12, 24, plot the aaytic solutio a the Mimo solutio. (You shoul have 4 separate plots, each with 2 graphs.) Be sure to vectorize the mimo operatio. Hit: Use equatios 12.9 a 12.3.

Lecture #3. Math tools covered today

Lecture #3. Math tools covered today Toay s Program:. Review of previous lecture. QM free particle a particle i a bo. 3. Priciple of spectral ecompositio. 4. Fourth Postulate Math tools covere toay Lecture #3. Lear how to solve separable