Two-step almost collocation methods for ordinary differential equations

Abstract

A new class of two-step Runge-Kutta methods for the numerical solution of ordinary differential equations is proposed. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. Local error estimation for these methods is also discussed.

Continuous two-step Runge–Kutta methods for ordinary differential equations

“…Additional results which confirm that continuous TSRK methods constructed in this paper preserve the order of convergence for stiff problems are given in [11].…”

“…As we aim for methods of order p = m, we impose the corresponding set of order conditions, stated in the following theorem (see [11,12]). …”

“…the characteristic polynomial of the stability matrix related to the class of methods (1.1). The general expression of the stability matrix of (1.1) has already been derived in [11,12] and takes the following form…”

“…However, for small values of ( = 10 −6 ) for which the van der Pol oscillator is stiff the Runge-Kutta-Gauss Table 5 Numerical results for Runge-Kutta-Gauss method of order p = 4 and stage order q = 2 for the Hires problem 6 .42 · 10 −7 2.25 10 1.47 · 10 −7 2.12 11 3.52 · 10 −8 2.06 12 8.62 · 10 −9 2.03 method exhibits order reduction phenomenon and its order of convergence drops to about p = 2 which corresponds to the stage order q = 2. This is not the case for TSRK method which preserves order of convergence p = q = 4, which leads to higher accuracy.…”

Multiplicative Adams Bashforth–Moulton methods

“…We can use (15) for E n (r) and E n+1 (r), in order to estimate the errors of multiplicative multi-step methods. As the rate of change between E n (r) and E n+1 (r) give the truncation errors of these methods for E n (r), we consider…”