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J ( r i + 3 r 2 ) , ^ = / ( * „ . 10) is called the ''optimal" two-stage Runge-Kutta process or Heun two-stage scheme. 11a) often called improved Euler or Heun method. 11b) and yn+\ =yn + hf{xn+Xj2, —(yn+yn+i)), (iii) a = c 2 =tf2i (semi-implicit). 12) It is now clear that there exist infinitely many two-stage Runge-Kutta processes of order two, depending on choice of the free parameter a. 2 The Explicit Two-Stage Process Example. 10) with an allowable error tolerance ε = IO"4. 13) is ^ ) = 1 + ïïk· (4 214) · We readily establish the following bounds: M = 10, L = 20, and \y2(xH9yn9h)\Z±ML2 = 4- * 10 · 202 .

E. e. , max I ίΛ I = 7 . t. y and JC, respectively. t. 8) lead to the following inequality: \en+l\ <(\+hL)\en\ + 7 \ / i = 0 , 1,.... , \en\ < ( 1 + / * L ) A M T + (l+hL)n \e0\ . hL For real z, 1 + z < e1. Therefore, {\+hL)n < enhL = eL{Xn~a). 12) hL A more illuminating bound for the global error can be derived by using more analytic properties of the IVP to obtain a sharper estimate for Γ. 6) yields tn+i=h(f(Xn+Qh,y(xn+Qh))-f(xn,y(xn))), 0<θ<1. We now add and subtract the quantity hf(xn+Qh, y(xn)) to the righthand side of the last equation, and, in addition, take the norm of both sides of the equation—the Lipschitz conditions o n / ( x , y)—in both variables to get \tn+l\

Y and JC, respectively. t. 8) lead to the following inequality: \en+l\ <(\+hL)\en\ + 7 \ / i = 0 , 1,.... , \en\ < ( 1 + / * L ) A M T + (l+hL)n \e0\ . hL For real z, 1 + z < e1. Therefore, {\+hL)n < enhL = eL{Xn~a). 12) hL A more illuminating bound for the global error can be derived by using more analytic properties of the IVP to obtain a sharper estimate for Γ. 6) yields tn+i=h(f(Xn+Qh,y(xn+Qh))-f(xn,y(xn))), 0<θ<1. We now add and subtract the quantity hf(xn+Qh, y(xn)) to the righthand side of the last equation, and, in addition, take the norm of both sides of the equation—the Lipschitz conditions o n / ( x , y)—in both variables to get \tn+l\

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