In this work a systematic procedure is implemented in order to minimise the computational cost of the Runge–Kutta–Munthe-Kaas (RKMK) class of Lie-group solvers. The process consists of the application of a linear transformation to the stages of the method and the analysis of a graded free Lie algebra to reduce the number of commutators involved. We consider here RKMK integration methods up to o...

A backward analysis of integration methods, whose numerical solution is a P-series, is presented. Such methods include Runge-Kutta methods, partitioned Runge-Kutta methods and Nystrr om methods. It is shown that the numerical solution can formally be interpreted as the exact solution of a perturbed diierential system whose right-hand side is again a P-series. The main result of this article is ...

Usually the straightforward generalization of explicit Runge{Kutta methods for ordinary diierential equations to half-explicit methods for diierential-algebraic systems of index 2 results in methods of order q 2 ((8]). The construction of higher order methods is simpliied substantially by a slight modiication of the method combined with an improved strategy for the computation of the algebraic ...

In this paper we consider the stability of variable step-size Runge-Kutta methods approximating bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to determine conditions on the maximum allowable step-size that guarantees the numerical solution of an asymptotically decaying time-dependent linear problem also de...

The equations governing the flow of an electrically conducting, incompressible viscous fluid over an infinite flat plate in the presence of a magnetic field are investigated using the homotopy perturbation method (HPM) with Padé approximants (PA) and 4 order Runge–Kutta method (4RKM). Approximate analytical and numerical solutions for the velocity field and heat transfer are obtained and compar...

In recent years, implicit stochastic Runge–Kutta (SRK) methods have been developed both for strong and weak approximations. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes such as simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of ...

This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay. ⎧⎨ ⎩ x′(t) +Bx(t) + Cx′(qt) +Dx(qt) = 0, t > 0, x(0) = x0, where B,C,D ∈ Cd×d, q ∈ (0, 1), and B is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient co...

Runge-Kutta and Adams methods are the most popular codes to solve numerically nonstiff ODEs. The Adams methods are useful to reduce the number of function calls, but they usually require more CPU time than the Runge-Kutta methods. In this work we develop a numerical study of a variable step length Adams implementation, which can only take preassigned step-size ratios. Our aim is the reduction o...

This work generalizes the additively partitioned Runge-Kutta methods by allowing for different stage values as arguments of different components of the right hand side. An order conditions theory is developed for the new family of generalized additive methods, and stability and monotonicity investigations are carried out. The paper discusses the construction and properties of implicit-explicit ...

0006-3207/$ see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.biocon.2013.01.010 ⇑ Corresponding author. Tel.: +1 608 781 6309; fax: +1 608 783 6066. E-mail addresses: [email protected] (W.E. Thogmartin), [email protected] (C.A. Sanders-Reed), [email protected] (P.C. McKann), [email protected] (L. Pruitt), [email protected] (R.A. King), [email protected] (M.C. Runge), r...