Question

1. We defined the Theta method: for 0 € [0, 1] Jtq+1 =t, + ; yn+1 = Yn + hof (tr; yn) + (1 - 0)f(tn+1 Yn+1)] Clearly, 0 - corresponds to the Trapezoidal rule, 0 = 1 to EEM, and 0 - o to IEM. Prove that the order of consistency of the Theta method is 1 for any

Please answer question #1 and write legibly. EEM is explicit euler method and IEM is implicit euler method

Answer 1

* let ux define the Theta method for 1 Anti-anth OE50117 Int Jeth Lof(enign) + (1-0) fltnt1 Ynto) det ut define the local truncation error for the o- method under the assumption that f is sufficiently smooth. Now the local truncation error food the o- method 18 det (loch) Po lich! = Iltats |-yang-h To f (tats Jl tatt)) (where yet) stands forson of ) 11-olf (th, Iltn) ] Arth) = fitnes ) - Iltn)-ho y' (tats 4 (1-0)y (th) Hence by expanding yltnes ) ylenth) & giltn+1) = yhtath) into the taylor senes around the point t= tn, we get four the local apronimation error melation - enth) = (1 - 0) 1h 2 y 11 (tn) 4 - 0 3 4 3 1 ltr Recall that the order of numerical to (44) method is defined by the local - truncation error: when ench) = ocepth) then method is called consistent of ourder p. This means both the Euleris method (for o=0 & 1 ) hap order of consistency 1 while for @= 1 ( trapezoidala the order of consistency is a two. dalne) 2 zoldal rule erroris =- (ant Olha) In Trapezoidal