High order stiffly stable composite multistep methods for numerical integration of stiff differential equations

Abstract

Let $$\dot y$$ =f(y,t) withy(t 0)=y 0 possess a solutiony(t) fort≧t 0. Sett n=t 0+nh, n=1, 2,.... Lety 0 denote the approximate solution ofy(t n) defined by the composite multistep method with $$\dot y_n $$ =f(y n ,t n ) andN=1, 2,.... It is conjectured that the method is stiffly stable with orderp=l for alll≧1 and shown to be so forl=1,..., 25. The method is intrinsically efficient in thatl future approximate solution values are established simultaneously in an iterative solution process with only one function evaluation per iteration for each of thel future time points. Step and order control are easily implemented, in that the approximate solution at only one past point appears in each component multistep formula of the method and in that the local truncation error for the first component multistep formula of the method is easily evaluated as $$T^{[l]} = \frac{h}{{t_{Nl} - t_{(N - 1)l - 1} }}\{ y_{Nl}^{PRED} - y_{Nl} \} ,$$ wherey Nl PRED denotes the value att Nl of the Lagrange interpolating polynomial passing through the pointsy (N−1)l+j att (N−1)l+j withj=−1, 0,...,l − 1.