16 Homework lecture 16

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1 Quees College, CUNY, Departmet of Computer Scece Numercal Methods CSCI 361 / 761 Fall 2018 Istructor: Dr. Sateesh Mae c Sateesh R. Mae Homework lecture 16 Please emal your soluto, as a fle attachmet, to Sateesh.Mae@qc.cuy.edu. Please submt oe zp archve wth all your fles t. 1. The zp archve should have ether of the ames CS361 or CS761: StudetId frst last CS361 hw16.zp StudetId frst last CS761 hw16.zp 2. The archve should cota oe text fle amed hw16.[txt/docx/pdf] ad oe cpp fle per questo amed Q1.cpp ad Q2.cpp etc. 3. Note that ot all homework assgmets may requre a text fle. 4. Note that ot all questos may requre a cpp fle. 1

2 16.1 Forward ad reverse tegrato: hgher order methods The followg dfferetal equato s a specal case of Rccat s equato dy dx x y We tegrate eq startg at x 0 0, wth the tal codto y 0 0. Suppose we tegrate eq forwards to x 1 ad the backwards to x 0. State the value of the exact soluto yx after tegratg back to x 0. I ths questo we shall umercally tegrate eq forward for steps to x 1, ad the backwards for steps to x 2 0. Employ a uform stepsze for the tegrato. 1. Use explct Euler, mdpot, trapezod, ad v Ruge Kutta fourth order RK4. 2. Compute the value of the umercal soluto y at x 1 ad fll the frst table of values below. 3. I all cases, compute the value of y to 3 decmal places d.p. 3 d.p. 3 d.p. 3 d.p d.p. 3 d.p. 3 d.p. 3 d.p d.p. 3 d.p. 3 d.p. 3 d.p. y Euler y mdpot y trapezod y RK4 Compute the value of the umercal soluto y 2 at x 2 0 ad fll the table of values below. Tabulate the values of y Euler 2 *** ote the powers of each case. ***, 3 y mdpot 2, 3 y trapezod 2 I all cases, state your aswers to 3 decmal places. 2 y2 Euler 3 y mdpot 2 3 y trapezod 2 5 y2 RK d.p. 3 d.p. 3 d.p. 3 d.p d.p. 3 d.p. 3 d.p. 3 d.p d.p. 3 d.p. 3 d.p. 3 d.p. ad v 5 y RK4 2. If you have doe your work correctly, the results each colum should be approxmately depedet of. Ca you expla why? You do t have to. See ext page for a explaato. 2

3 The explct Euler method s frst order, hece ts local trucato error s Oh 2 O1/ The umercal errors add up accumulate magtude o both the forward ad reverse tegratos. 2. Hece the overall umercal error after 2 tegrato steps s Oh 2 O1/. The mdpot ad trapezod methods are secod order, hece the local trucato error both cases s Oh 3 O1/ However otce that the overall error the value of y 2 s also Oh 3 O1/ The overall error has NOT grow to Oh 3 O1/ Ths s because both tegrators have some symmetry, f the drecto of tegrato s reversed, ad the umercal errors o the reverse tegrato partally cacel the errors accumulated o the forward tegrato. 4. Ruge Kutta RK4 also dsplays the same good cacellato property. The local trucato error s Oh 5 O1/ 5 ad the overall error the value of y 2 s also Oh 5 O1/ 5. Ths s a good feature for a terator to have. 1. ote that the forward backward symmetry s ot exact, for the mdpot, trapezod ad RK4 methods. 2. It s a approxmate symmetry. 3. Hece the cacellato s partal, ot complete. 4. evertheless, t s a good feature. 5. There are other tegrators where the forward backward symmetry s exact. 6. Typcally, for such tegrators, the dfferetal equatos must have a specal structure. 3

4 16.2 Area eclosed by curve Let α > 0 ad β > 0 be postve costat real umbers. You are gve the followg dfferetal equato: The tal codto s A 0 at x 0. da dx 4 1 xα 1/β Show that the value of A1 s gve by the followg tegral: A x α 1/β dx Show that the value of A1 s the area eclosed by the followg closed curve: Set α 0.7 ad β 1.2. x α + y β Itegrate eq usg steps from x 0 0 to to x Employ a uform stepsze h 1/. 2. Use explct Euler, mdpot, trapezod, ad v Ruge Kutta fourth order RK4. 3. Defe A as the umercal value of A1. 4. Compute the value of the umercal soluto A at x 1 ad fll the table of values below. 5. I all cases, compute the values of A to 4 decmal places d.p. 4 d.p. 4 d.p. 4 d.p d.p. 4 d.p. 4 d.p. 4 d.p d.p. 4 d.p. 4 d.p. 4 d.p. A Euler A mdpot A trapezod A RK4 4

5 16.3 Auxlary varables The pedulum equato ca be wrtte the form d 2 θ + k s θ dt2 The depedet varable s the tme t ad k > 0 s a real postve costat. Itroduce a auxlary varable v, ad wrte dθ dt v, dv k s θ. dt b a Defe a parameter E as follows E v2 2 + k1 cos θ Use eqs a ad b to show the followg: de dt We call E a varat. Its value does ot chage alog a trajectory the θ, v space. I fact, E s the eergy of the pedulum. Defe a two compoet vector y va Defe a two compoet vector f f 1, f 2 va y θ, v f 1 t, θ, v v, f 2 t, θ, v k s θ The formal equato s dy/dt f. Use eqs a ad b to show the followg: dy dt d dt θ v f1 t, θ, v f f 2 t, θ, v 5

6 The tal codtos are θ 0 ad v 2 at t t 0 0. Set k 1. Calculate the value of E eq for these tal codtos. Itegrate eq forward tme usg a stepsze of h Use explct Euler, mdpot, trapezod, ad v Ruge Kutta fourth order RK4. 2. The C++ fucto code s show o the ext page. 3. For each method, compute the umercal value of the followg quatty: D v2 2 + k1 cos θ E Fll the table below wth the values of D Euler 10 5 ad v D RK *** Note the powers of 10. *** D trapezod 6. I all cases, state the result to 2 decmal places., D mdpot 10 5, D Euler D mdpot 10 5 D trapezod 10 5 D RK d.p. 2 d.p. 2 d.p. 2 d.p d.p. 2 d.p. 2 d.p. 2 d.p d.p. 2 d.p. 2 d.p. 2 d.p d.p. 2 d.p. 2 d.p. 2 d.p. Optoal 1. For 1000, plot separate graphs of D agast θ, 0, 1,...,, computed usg explct Euler, mdpot, trapezod, ad v Ruge Kutta fourth order RK4. 2. For 1000, plot separate graphs of the soluto the θ, v plae,.e. plot the pots θ, v, 0, 1,...,, computed usg explct Euler, mdpot, trapezod, ad v Ruge Kutta fourth order RK4. Let us aalyze the detals of varous tegrato schemes. See ext pages. 6

7 C++ fucto code All the tegrato schemes call the followg C++ fucto. It says x stead of t : t ft m, double x, cost std::vector<double> & y, std::vector<double> & g { // value of k s hard-wred for smplcty cost double k 1.0; // frst compoet "f1t,theta,v v" g[0] y[1]; // secod compoet "f2t,theta,v -k*stheta" g[1] -k*sy[0]; } retur 0; All the tegrato schemes ca be called wth the followg puts: // m 2 // x t_ // h step sze // vector array y_ theta, v_ // vector array y_out theta, v_{+1} t Euler_explctt m, double x, double h, std::vector<double> & y_, std::vector<double> & y_out; t mdpott m, double x, double h, std::vector<double> & y_, std::vector<double> & y_out; t trapezodt m, double x, double h, std::vector<double> & y_, std::vector<double> & y_out; t RK4t m, double x, double h, std::vector<double> & y_, std::vector<double> & y_out; 7

8 Explct Euler method I ths scheme, we have y +1 y + h ft, y I matrx otato ths meas θ v +1 I vector compoet otato ths meas Frst step 0. θ + h v f1 t, θ, v f 2 t, θ, v θ +1 θ + h f 1 t, θ, v θ + h v, a v +1 v + h f 2 t, θ, v v hk s θ b 1. The t 0 0, θ 0 0 ad v Hece Secod step 1. θ 1 θ 0 + h f 1 t 0, θ 0, v 0 θ 0 + h v 0 2 h, a v 1 v 0 + h f 2 t 0, θ 0, v 0 v 0 hk s θ b 1. The t 1 h, θ 1 2 h ad v Hece θ 2 θ 1 + h f 1 t 1, θ 1, v 1 θ 1 + h v h, a v 2 v 1 + h f 2 t 1, θ 1, v 1 v 1 hk s θ 1 2 hk s 2 h b 8

9 Mdpot method I ths scheme, we have g 1 ft, y, g 2 ft h, y hg 1, y +1 y + h g a b c I matrx otato ths meas gθ g v 1 f1 t, θ, v v f 2 t, θ, v k s θ Next defe y hg 1 θtmp v tmp θ v + h 2 gθ g v 1 θ hv v 1 2 hk s θ Next Frst step 0. gθ g v 2 f1 t h, θ tmp, v tmp v f 2 t h, θ tmp tmp, v tmp k sθ tmp The t 0 0, θ 0 0 ad v Hece g θ 1 f 1 t 0, θ 0, v 0 v 0 2, a g v 1 f 2 t 0, θ 0, v 0 k s θ b 3. The θ tmp θ h g θ 1 h/ 2, a v tmp v h g v b 4. Next g θ 2 f 1 t h, θ tmp, v tmp v tmp 2, a g v 2 f 2 t h, θ tmp, v tmp k sθ tmp k sh/ b 5. The θ 1 θ 0 + h g θ 2 2 h, a v 1 v 0 + h g v 2 2 hk sh/ b Secod step Yuck. 9

10 Trapezod method I ths scheme, we have g 1 ft, y, g 2 ft + h, y + hg 1, y +1 y + h 2 g 1 + g a b c I matrx otato ths meas Next defe Next Frst step 0. y + hg 1 gθ g v 2 gθ g v 1 θtmp v tmp f1 t, θ, v v f 2 t, θ, v k s θ 1. The t 0 0, θ 0 0 ad v Hece 3. The 4. Next θ v + h gθ f1 t + h, θ tmp, v tmp v tmp f 2 t + h, θ tmp, v tmp k sθ tmp g v 1 θ + hv v hk s θ g θ 1 f 1 t 0, θ 0, v 0 v 0 2, a g v 1 f 2 t 0, θ 0, v 0 k s θ b θ tmp θ 0 + h g θ 1 2 h, a v tmp v 0 + h g v b g θ 2 f 1 t 0 + h, θ tmp, v tmp v tmp 2, a g v 2 f 2 t 0 + h, θ tmp, v tmp k sθ tmp k s 2 h b 5. The θ 1 θ 0 + h [ ] gθ 1 + g θ 2 2 v 1 v 0 + h [ ] gv 1 + g v 2 2 Secod step 1. 2 h, a 2 hk 2 s 2 h b 1. Yuck. 10

11 Ruge Kutta RK4 I ths scheme, we have g 1 ft, y, g 2 ft h, y hg 1, g 3 ft h, y hg 2, g 4 ft + h, y + hg 3, y +1 y + h 6 g 1 + 2g 2 + 2g 3 + g a b c d e Frst step The t 0 0, θ 0 0 ad v Hece 3. The 4. Next g θ 1 f 1 t 0, θ 0, v 0 v 0 2, a g v 1 f 2 t 0, θ 0, v 0 k s θ b θ tmp θ h g θ 1 h/ 2, a v tmp v h g v b g θ 2 f 1 t h, θ tmp, v tmp v tmp 2, a g v 2 f 2 t h, θ tmp, v tmp k sθ tmp k sh/ b 5. The defe 6. Next θ tmp θ h g θ 2 h/ 2, a ṽ tmp v h g v 2 2 hk 2 sh/ b g θ 3 f 1 t h, θ tmp, ṽ tmp ṽ tmp 2 hk 2 sh/ 2, a g v 3 f 2 t h, θ tmp, ṽ tmp k s θ tmp k sh/ b 7. The defe ˆθ tmp θ 0 + h g θ 3 2 h h2 k 2 sh/ 2, a ˆv tmp v 0 + h g v 3 2 hk sh/ b 11

12 8. Next 9. The g θ 4 f 1 t 0 + h, ˆθ tmp, ˆv tmp ˆv tmp 2 hk sh/ 2, a g v 4 f 2 t 0 + h, ˆθ tmp, ˆv tmp k sˆθ tmp k s 2 h h2 k 2 sh/ 2. Secod step Oh dear b θ 1 θ 0 + h [ ] gθ 1 + 2g θ 2 + 2g θ 3 + g θ 4, a v 1 v 0 + h [ ] gv 1 + 2g v 2 + 2g v 3 + g v b 12