Maximum polynomial degree Nordsieck–Gear (k, p) methods: Existence, stability, consistency, refinement, convergence and computational examples

摘要

This paper revisits and extends the work that we started in 1972 on maximum polynomial degree (m.p.d) (k, p) Nordsieck–Gear methods for the numerical solution of the non-stiff ordinary differential equations initial value problem:y(p)(x)=f(x,y,y(1),y(2),…,y(p-1)),y(0)=y0,y(j)(0)=y0(j),j=1,2,…,p-1,f ∈ Ck+2(N), N a neighborhood of the graph (x,y¯(x),y¯(1)(x),…,y¯(p-1)(x)),x∈[0,X],y¯(x),y¯(1)(x),y¯(2)(x),…,y¯(p-1)(x) the exact solution values of y and its derivatives up to order p − 1.In the notation (k, p) k is the number of scaled derivatives that are retained by the method, p is the order of the differential equation and k > p ⩾ 1. These methods belong to the class of linear multi-value predictor–corrector methods. They are fixed order, variable step and admit of accurate self-starting algorithms.In our first paper we proved that, given exact starting values for the p + 1, p + 2, … , k scaled derivative starting values, a vector of p Nordsieck–Gear (k, p) corrector–predictor coefficients must be recursively computed to preserve m.p.d k + 1. Computational experiment showed that the sequence of coefficient vectors was well-defined in the (k, 1) case for k = 2, 3, 4, 5. This led to the two-part conjecture: (1) The sequence of coefficient vectors is well-defined for all (k, p) and (2) These coefficient vectors cycle to their fixed values at most k − p + 1 steps.Our second paper dealt with the construction of a starting method which yields starting scaled derivatives of order p + 1, p + 2, … , k, all accurate to O(hk+2), with exactly (k − p + 1)(k − p + 2)/2 + 2 evaluations of the derivative function f.In this paper we begin by proving the first conjecture and a slightly stronger version of the second. With these results in hand we then prove convergence in the (k, 1) method. It is shown that the starting algorithm’s guaranteed O(hk+2) starting error for the (k, 1) method can be preserved globally for sufficiently small step-size h by augmenting two corrector–predictor evaluations with a stability-preserving, computationally efficient refinement.We have coded a (5, 1) Fortran 77 algorithm, DIFFEQ, to implement, test, and confirm the underlying theory. We compare its computational efficiency with the variable order, variable step Adams–Gear method contained in the 1995 edition of the International Mathematical and Statistical Library (IMSL).