Numerical Methods for Partial Differential Equations

Transcription

1 Numerical Methods for Partial Differetial Equatios CAAM 452 Sprig 2005 Lecture 9 Istructor: Tim Warburto

2 Today Chage of pla I will go through i detail how to solve homework 3 step by step.

3 Q1) Build a fiite-differece solver for Homework 3 Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio u t Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. u = c x ) 2 π x) u x e x [ ] 4cos,0 =, 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

4 Q1a Use Cash-Karp RK for Time- Steppig We will use a vector versio of Cash-Karp: 1 k = dtf u ) 2 1 k = dtf u + b ) 21k k = dtf u + b ) 31k + b32k k = dtf u + b ) 41k + b42k + b43k k = dtf u + b ) 51k + b52k + b53k + b54k k = dtf u + b61k + b62k + b63k + b64k + b65k u = u + c k + c k + c k + c k + c k + c k ) Where f is a vector valued fuctio i our case it will be a liear operator actig o a vector argumet.

5 Where the b,c coefficiets are c= [ 37/ / / /1771] /5 0 3/40 9/40 0 b = 3/10-9/10 6/5 0-11/54 5/2-70/27 35/ / / / / / Provided i cashkarp.m o the web site. Origial paper at:

6 Q1) Build a fiite-differece solver for Homework 3 u t Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio ) 4cos Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. u = c x 2 π x) [ ] u x,0 = e, x 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

7 Recall: 4 th Order Cetral Differece Scheme The fourth order cetral differece derivative acts o ay vector ad gives the followig value for each etry of a result vector: 1 δ 4v ) = vm 2 8vm 1 8vm 1 vm 2 ) m dx Where the icremets o the idexig is doe modulo M. Sice we will use this operator repeatedly I made a fuctio

8 Code Each time step cosists of evaluatig the 6 itermediate vectors k1,k2,..,k6 Ad piecig them together. Notice here I set up u at the start as a vector, ad delta4 accepts a vector ad dx) as argumets ad returs a vector. i.e. each of k1,k2,..,k6 is a vector.

9 Alterate Code I could also have used loops Note: u is a row vector of legth M so the k s are a matrix of size 6 x M

10 Q1) Build a fiite-differece solver for Homework 3 u t Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio ) 4cos Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. u = c x 2 π x) [ ] u x,0 = e, x 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

11 Cash-Karp The Cash-Karp Ruge-Kutta itegrator i our case is: k 1 = ) dtf u ) k = dtf u + b k ) k = dtf u + b k + b k ) k = dtf u + b k + b k + b k ) k = dtf u + b k + b k + b k + b k ) k = dtf u + b k + b k + b k + b k + b k u = u + c k + c k + c k + c k + c k + c k Notice, we do ot eed the time compoet o the right had side, because our fiite-differece right had side is idepedet of time.

12 Liear Stability Aalysis We achieve the liear stability aalysis by assumig f is liear i u: f u) = µ u k 1 = µ dtu ) k = µ dt u + b k ) k = µ dt u + b k + b k ) k = µ dt u + b k + b k + b k ) k = µ dt u + b k + b k + b k + b k ) k = µ dt u + b k + b k + b k + b k + b k u = u + c k + c k + c k + c k + c k + c k

13 cot We replace mu*dt with u k 1 = νu ) k = ν u + b k ) k = ν u + b k + b k ) k = ν u + b k + b k + b k ) k = ν u + b k + b k + b k + b k ) k = ν u + b k + b k + b k + b k + b k u = u + c k + c k + c k + c k + c k + c k

14 cot We wish to remove all the itermediate variables ad express the scheme i a oe-step form like: u Where the multiplier fuctio piu) is a 6 th order polyomial i u. It is certaily possible to fid all the coefficiets of this polyomial by had but a little messy. We ca take a short cut assumig Cash-Karp is actually a 5 th order scheme as claimed we kow that the first 6 terms of the polyomial multiplier must be the first 6 terms of the Taylor series for ) + 1 = π ν u e ν

15 cot So we kow that: π ν = 1+ ν + ν + ν + ν + ν + Cν 1! 2! 3! 4! 5! ) Where C is to be determied. However, lookig at the defiitio of the stages we see that there is oly oe cotributio to the 6 th order term: 6 c b b b b b ν So π ν = 1+ ν + ν + ν + ν + ν + c b b b b b ν 1! 2! 3! 4! 5! )

16 Stability Coditio We eed to plot the stability regio, so we determie the margi of stability by fidig u such that π ν ) = 1+ ν + ν + ν + ν + ν + c6b65b 54b43b32 b21ν = 1 1! 2! 3! 4! 5! The curve of poits i the root) complex plae with i magitude 1 ca be parameterized i theta by e θ So for each theta i [0,2pi) we seek the correspodig u such that: i π ν = 1+ ν + ν + ν + ν + ν + c b b b b b ν = 1! 2! 3! 4! 5! ) e θ

17 Matlab roots Commad The matlab roots commad ca be used to fid roots of a polyomial. If the polyomial is say: ) π x = a + a x + a x The costruct a vector of coefficiets highest order first): a = [ a, a, a ] Ad ivoke: rootsa) A vector of the roots if ay) of the polyomial will be retured.

18 Adaptig RKabstab.m So I took the routie from the class web page ad modified it slightly I hard coded it to use the multiplier polyomial for the Cash-Karp I chaged the plottig to use a scatter plot

19 Absolute Stability for Cash-Karp This is a classic picture. What is iterestig is that it does ot covicigly cover the imagiary axis!.

20 O The Imagiary Axis Here I used Matlab s polyval to evaluate the multiplier polyomial for u o the imagiary axis. Coclusio the absolute stability regio is just to the left of the imagiary axis ot a big issue here)

21 Q1) Build a fiite-differece solver for Homework 3 u t Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio ) 4cos Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. u = c x 2 π x) [ ] u x,0 = e, x 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

22 Q1d) Boud the spectral radius of the derivative operator All the eigevalues of the 4 th order cetral differece are o the imagiary axis, with values recall from Lecture 6): ic λ = θ θ m 6dx m m 8si ) si 2 )) So the gotcha is that there is o dt such that the eigevalues*dt are all exactly iside the stability regio. However, we will estimate the largest eigevalue ad make a assumptio..

23 Q1d) cotiued ic λ θ ) = 8si θ ) si 2θ )) 6dx dλ ic θ = θ θ dθ 6dx critical poits for θ which satisfy: ) ) 8cos ) 2cos 2 ) θ ) θ ) 0 = 8cos 2cos 2 2 θ ) θ ) ) =8cos 2 2cos 1 θ ) θ ) 2 =-4cos + 8cos ± i. e. cos θ ) = = i e θ 3 2 c dx -1.. = cos 1- ) m λ We ca estimate the largest magitude eigevalue directly

24 Q1) Build a fiite-differece solver for Homework 3 u t Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio ) 4cos Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. u = c x 2 π x) [ ] u x,0 = e, x 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

25 Q1e) Sice we are oly itegratig for 8 periods let s choose a reasoable maximum u <=1 So for close to absolute liear) stability Implies that we ca choose: i.e. the time step restrictio is: We ca further reduce the right had side of the iequality to reduce margial growth. cdt dx λ m dx dt c c dx

26 Q1) Build a fiite-differece solver for Homework 3 Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio u t Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. ) u = c x 2 π x) u x e x [ ] 4cos,0 =, 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

27 Q1f: Test Rus I already set up this iitial coditio for the web demos: I modified the testrig.m file to call the cetraldifferece4ckrk54.m file T=4 It looks like reasoable 4 th order covergece.. We examie the error ratios to be more certai

28 Q1) Build a fiite-differece solver for Homework 3 Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio u t ) Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. u = c x 2 π x) u x e x [ ] 4cos,0 =, 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

29 Q1g) Estimate Order of Accuracy of Solutio M Max error at t= Order of accuracy Cool asymptotically it looks like a fourth order code i space)

30 Q1) Build a fiite-differece solver for Homework 3 Q1a) use your Cash-Karp Ruge-Kutta time itegrator from HW2 for time steppig Q1b) use the 4 th order cetral differece i space periodic domai) Q1c) perform a stability aalysis for the time-steppig based o visual ispectio of the CK R-K stability regio cotaiig the imagiary axis) Q1d) boud the spectral radius of the spatial operator Q1e) choose a dt well i the stability regio u t Q1f) perform four rus with iitial coditio use M=20,40,80,160) ad compute maximum error at t=8 Q1g) estimate the accuracy order of the solutio. ) u = c x 2 π x) u x e x [ ] 4cos,0 =, 0,2 Q1h) extra credit: perform adaptive time-steppig to keep the local trucatio error from time steppig bouded by a tolerace.

31 Q1h) Estimatig the Local Trucatio Error Give the k vectors we ca also costruct a lower order 4 th order i time) approximatio to the updated solutio: 1 * 1 * 2 * 3 * 4 * 5 * 6 u + = u + c1 k + c2k + c3k + c4k + c5k + c6k The we ca ball park what the time step should be: dt rk 4 tol u = dt mi u u est rk 5 rk 4 0.2

32 Q1h) Code Modify last time step to lad o EdTime Form alterative embedded fourth order approximatio Estimate reasoable dt Check to see if we should reduce time step.

33 Q1h) Testig I started with a fairly large time step ad let the code adjust itself M Max error at t= Order of accuracy These are pretty much uchaged from before.

34 Q1h) Adaptive Time Steppig started with dt=100*dx) Q) Why does it take more time steps for the Lower resolutio rus?

35 Note o the Cash-Karp RK Check out: There is a iterestig treatmet of RK schemes, with the idea of parameterizig embedded RK schemes +1 stage, th order) by choosig the coefficiet i the multiplier directly. Icreasig this parameter a little will make sure that at least part of the imagiary axis is iside the absolute stability regio.