EFFICIENT HIGH ORDER SECOND DERIVATIVE METHODS FOR IVPS

Second derivative general linear methods (SGLMs) for the numerical solution ofy′(x) = f(y(x)), x 2 [x0, x], y(x0) = y0,where f : R(m) → R(m) and m is the dimensionality of the system, are multistage-multivalue methods given byY [n] = h(A ⨂ I(m)) F(Y [n]) + h(2) (Ā ⨂ I(m)) G(Y [n]) + (U ⨂ I(m)) y[n−1],Y [n] = h(B ⨂ I(m)) F(Y [n]) + h(2) (B⨂ I(m)) G(Y [n]) + (U ⨂ I(m)) y[n−1],where h is the stepsize, A, ĀЄ Rs×s, U Є R(s×r), B, B Є R(r×s) and V Є R(r×r) andnotation ⨂ is the Kronecker product.The main properties of SGLMs as pre–consistency, consistency, zero-stability to-gether with types of these methods have been discussed in [1]. Construction of SGLMstogether with some order barriers for different types of these methods have been inves-tigated in [2, 3]. Construction of these methods of high orders in which the number ofinternal stages is equal to the order leads to solve the nonlinear systems of equationsfor satisfying RKS property which is impossible. In this paper, we construct A–stableSGLMs of high order and stage order with fewer stages.

Keywords:

Second derivative methods , General linear methods , RK stability