Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.
A numerical method applied to the differential equation yr=Ly can lead to a solution of the form y(x, + nh) =Azn, where z and w = Lh satisfy an equation P(w, z ) =O. In general P(w, z ) is a polynomial in both w and z, of degree M in w and N in z. Existing multistep and Rung* Kutta methods correspond to the cases M = l and N = l respectively. New methods are found by taking M 2 2 , N 22. The approach here is first to find a suitable polynomial P ( w , z ) with the desired stability properties, and then to find a process which leads to this polynomial. Third-and fourth-order -4-and L-stable processes are given of the semi-explicit and linearly implicit types.
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