A new third order convergent numerical solver for continuous dynamical systems

Abstract

Continuous dynamical systems are attempted to solve via proposed third order convergent numerical solver. The solver is shown to possess third order convergence on the basis of containing O(h4) term in the leading coefficient of the local truncation error which has further been employed for the local and global error bounds whereas its increment function is found to be Lipchitz continuous. Consistency of the solver has extensively been discussed using Taylor’s series expansion. The necessary and sufficient conditions for the solver to be stable have been proved accompanying the linear stability function obtained via Dahlquist test problem. In order to illustrate the performance of the solver, few scalar and vector-valued dynamical systems of both linear and nonlinear nature have numerically been solved in comparison to two well-known numerical solvers having same order of convergence as that of the proposed solver. Whereupon, the proposed solver is found to contain the smallest errors. The analysis of comparison is based upon three kinds of errors; namely, maximum absolute global relative error, absolute relative error determined at the final nodal point along the integration interval and the ℓ2-error norm. MATLAB software having version 9.3.0.713579 (R2017b), using Intel(R) Core(TM) i3-4500U processor running on 1.70 GHz has been used.