Recently Ch. Lubich proved convergence results for Runge-Kutta methods applied to stii mechanical systems. The present paper discusses the new ideas necessary to extend these results to general linear methods, in particular BDF and multistep Runge-Kutta methods. Stii mechanical systems arise in the modelling of mechanical systems containing strong springs and (or) elastic joints. A typical exam...

The study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system, whose discretisation is related to reverse accumulation in automatic differentiation. Similar adjoint systems arise in optimal control and other areas, including classical Mechanics. Adjoint systems are introduced in...

The study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system, whose discretization is related to reverse accumulation in automatic differentiation. Similar adjoint systems arise in optimal control and other areas, including classical mechanics. Adjoint systems are introduced in...

in this article, we present a fractional order hiv-1 infectionmodel of cd4+ t-cell. we analyze the effect of the changing the averagenumber of the viral particle n with initial conditions of the presentedmodel. the laplace adomian decomposition method is applying to checkthe analytical solution of the problem. we obtain the solutions of thefractional order hiv-1 model in the form of infinite se...

The Interaction Picture (IP) method is a valuable alternative to Split-step methods for solving certain types of partial differential equations such as the nonlinear Schrödinger equation or the Gross-Pitaevskii equation. Although very similar to the Symmetric Split-step (SS) method in its inner computational structure, the IP method results from a change of unknown and therefore do not involve ...

The problem of thermal diffusion and diffusion thermo effects on thermosolutal Marangoni convection flow of an electrically conducting fluid over a permeable surface is investigated. Using appropriate similarity transformations, the governing system of partial differential equation is transformed to a set of nonlinear ordinary differential equations, then solved numerically using the Runge-Kutt...

In this paper, we introduce a method for improving the embedded Runge-Kutta-Fehlberg4(5) method. At each integration step, the proposed method is comprised of two equations for the solution and the error, respectively. These solution and error are obtained by solving an initial value problem whose solution has the information of the error at each integration step. The constructed algorithm cont...

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge-Kutta time integrators, with the aim of deriving optimal Runge-Kutta schemes for wave propagation applications. We review relevant Runge-Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user...

Implicit integration schemes for ODEs, such as Runge-Kutta and Runge-Kutta-Nyström methods, are widely used in mathematics and engineering to numerically solve ordinary differential equations. Every integration method requires one to choose a step-size, h, for the integration. If h is too large or too small the efficiency of an implicit scheme is relatively low. As every implicit integration sc...

A nine-stage multi-derivative Runge–Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. The method uses y′ and higher derivatives y(2) to y(6) as in Taylor methods and is combined with a 9-stage Runge–Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor e...