Numerical analysis of dynamic system(Master’s degree 2020)(All calculation results or figures obtained by computer must beattached with the program codes)(2 pages in total. Test paper and answer paper shall be submittedtogetherú)
1. Consider backward differentiation formula (BDF).
(a) Prove that thek-step BDF method is stable if and only ifk≤6 holds.
(b) Prove that thek-step BDF method isA-stable ifk= 1,2.
(c) Draw the regions of absolute stability of thek-step BDF withk= 1,2,3,4,5,6, respectively.
2. Try to construct 3-stage Runge Kutta method (A, b, c) of Guasstype.(a) Discuss the existence and uniqueness of the solution pro-duced by the method when it is applied toy′(t) =f(y(t)),wherefsatisfies the Lipschitz condition.(b) Verify that the method is of order 6.(c) Prove that the method is symplectic.
3. Consider the initial value problem{dy(t)dt=Ay(t) +φ(t), t≥0,y(0) = (0,0,···,0,0)T∈Rm−1,(0.1)whereA=m2−2 11−2 1………1−2 11−2∈R(m−1)×(m−1),φ(t) =m2(1,0,···,0,−1)T∈Rm−1.(a) Find the all eigenvalues ofA.(b) Givenm= 10ßsolve (0.1) by using Euler method with step-sizeh= 0.0045,0.01, respectively. Observe the numericalresults att= 1 and present your comments.(c) Givenm= 10ßsolve (0.1) by using implicit midpoint methodwith stepsizeh= 0.0045,0.01, respectively. Compare with1
the numerical results of case (b), find out their differencesand present your explanation.(d) Givenh= 0.001ßsolve (0.1) withm= 10,100 by usingEuler method, respectively. Observe the numerical resultsatt= 1 and present your comments.
4. Consider the initial value problemdudt=u(v−2) :=a(u, v),dvdt=v(1−u) :=b(u, v),u(0) =u0, v(0) =v0,(0.2)wheret >0. Let stepsizeh= 0.1. Solve (0.2) by using thefollowing three methods:(a) Explicit Euler method with (u0, v0) = (1,1)∂(b) Implicit Euler method with (u0, v0) = (4,4)∂(c)un+1=un+ha(un, vn+1),vn+1=vn+hb(un, vn+1),(u0, v0) = (5,2).(0.3)Draw figures of the numerical results with respect to the abovethree methods in phase space, respectively, wheret∈[0,10000].Observe the figures and explain the phenomena observed.