Question

(e) Consider the Runge-Kutta method in solving the following first order ODE: dy First, using Taylor series expansion, we have the following approximation of y evaluated at the time step n+1 as a function of y at the time step n: where h is the size of the time step. The fourth order Runge-Kutta method assumes the following form where the following approximations can be made at various iterations: )sh+รู้: ,f(t.ta, ),. Note that the first term is evaluated at the initial point, the middle two terms are evaluated at mid-point, and the last term is evaluated at the end point of the time step. Find the constants a, b, c, and d in the fourth order Runge-Kutta method such that the error term in the Runge-Kutta method is proportional to O(h) (7 marks)

Answer 1

Censtaert the e kuila methoal in Batuin the felleaitrg Arot Ftst, cin Taylore Series expavvion, cve. have the foll wing appreto- 1imation of evalulateal at the time step Dt1 a a feanction of at the time ,tepn cohere h is the vize of the time tep The fourth orrolerc R-K metbod a sumes th following form aha-a.th[af(inth) tbtce,tde , kn) t crent cohere the fooruwing oprozimation can Le kna) tdtunt.m] made varc iou ao Sobrtitudtny al these values in Ca) e have 3! 12 Compasrting this coth the oylore erte) have 1 a-b+ C+d =

and ad=1 Thus ,巳have ; 6-2-1 2 6- 3-3-1 = i一字- すで The herth orderて ,'' Rouge-kutta method. Agivenky nta n