DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science

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1 DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Matematics and Computer Science. ANSWERS OF THE TEST NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS (WI3097 TU) Tuesday January 9 008, 9:00-:00 a) In te general formulation, f(t n+, w n+ ) is replaced by λw n+ : w n+ = w n + f(t n+, w n+ ), w n+ = w n + λw n+. Solving for w n+, we find w n+ = λ w n. It follows, by definition, tat te amplification factor equals Q(λ) = b) For a λ wit a negative real part, we ave λ. Q(λ) = Re(λ) iim(λ) = ( Re(λ)) + (Im(λ)) Since Re(λ)) 0 it appears tat Re(λ), so Q(λ) = ( Re(λ)) + (Im(λ)) independent of. Hence, te BE metod is unconditionally stable. c) Te local truncation error is defined as τ n+ = y n+ w n+. () Te exact solution of te test equation y = λy can be written as y n e λ(t tn) ; ence at t = t n + we ave y n+ = e λ y n.

2 Te quantity w n+ is defined as te numerical solution on te interval (t n, t n+ ), starting from te exact value y n. So, for te test equation : w n+ = y n + λ w n+, suc tat (compare (a)) w n+ = y λ n. Now insert bot expressions into te definition (): τ n+ = eλ λ y n, and replace e λ and by teir expansions + λ + λ λ +..., respectively +λ+ λ Te first two terms of te expansions cancel and we are left wit τ n+ = wic proves tat BE is O(). λ +... ( λ +...) = O(), d) Calling x = y and x = y, te first differential equation follows directly: x = x. Note tat x = y, substituting y = 000.5y 500y = 000.5x 500x yields te second equation: x = 500x 000.5x. Using matrix-vector notation, te two equations are compiled as x = Ax, were [ A = To find te eigenvalues, te determinant of [ λ λ is put equal to 0. Tis leads to te quadratic equation λ λ = 0 wit roots λ = 000 and λ = 0.5. Note tat bot eigenvalues are negative and tat λ is muc larger tan λ. e) For an O( p )- metod te error at time t is estimated by te general formula ((6.5) of te lecture notes) y(t) w(t, ) w(t, ) w(t, ). p Using p = and = 0.06, application of tis formula to te given table values yields ]. ], y w (3.6, 0.03) w (3.6, 0.03) w (3.6, 0.06) = as te BE-error for te second component (te derivative of te solution) at t = 3.6. Tis is well witin te given tolerance of

3 f) Te stability condition of Forward Euler, applied to y = λy, reads < λ. To apply tis condition to te system derived in (d), we ave to substitute its (in absolute value largest) eigenvalue -000 for λ. It follows tat as to satisfy te condition < g) For a step size wic is close to its maximal value 0.00, Forward Euler produces a result wit a given error of 0.000, far less tan te required accuracy of From te point of view of efficiency we would like to increase te step size but tat is impossible because of stability requirements. Because bot Euler metods are O() teir accuracy is comparable, but Backward Euler is unconditionally stable and ence, te step size can be increased at will. Te error estimate in (e) as sown tat a step size of 0.03, at least 5 times larger tan te maximal stable step size of Euler Forward, is sufficient to meet te required accuracy. So, Euler Backward is te most suitable metod.. (a) A fixed point p satisfies te equation p = g(p). Substitution gives: p = p + p. Rewriting tis expression gives: 0 = p p = p = p = ±. On te oter and f(p) = 0 gives p + p = 0 p + p = p = + p p = p = ±. So bot expressions leads to te same solutions. 3

4 Te fixed point iteration is defined by: p i+ = g(p i ). Starting wit p 0 = one obtains: p = 0.875, p = 0.99, p 3 =. (b) For te convergence two conditions sould be satisfied: g(p) [, ] for all p [, ]. g (p) k < for all p [, ]. Since g(p) = p + p, te derivative is g (p) = p. Note tat g (p) 0 for all p [, ]. Tis implies tat = g( ) g(p) g() =, so te first condition olds. For te second condition we note tat g (p) = p = k < for all p [, ], so te second conditions is also satisfied, wic implies tat te fixed point iteration is convergent for all p 0 [, ]. (c) Since g (p) = p it follows tat so te metod is divergent. g ( ) = = >, (d) Grapically te Newton-Rapson metod is given in Figure. Te tangent in f(x) tangent p 0 p 0 f(p 0 ) Figure : Te Newton-Rapson metod (p 0, f(p 0 )) is given by: l(x) = f(p 0 ) + (x p 0 )f (p 0 ).

5 Taking l(p ) = 0 leads to Rewriting gives p = p 0 f(p 0) f (p 0 ). (e) Starting wit p 0 = f(p 0 ) + (p p 0 )f (p 0 ) = 0. we note tat f(p) = f (p) = p + p, p ( + p ). Substituting tis into te formula gives p = ( ) + = (+ ) (f) Note tat ˆp i+ p i+ = ˆp i ˆf(ˆp i ) ˆf (ˆp i ) (p i f(p i) f (p i ) ). From te assumptions ˆp i = p i and ˆf (p i ) = f (p i ) it follows tat ˆp i+ p i+ ˆf(p i ) f(p i ) f (p i ) ǫ f (p i ) ǫ, since f (p) = p ( + p ) ( + ) =. 5