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...

This study utilised positive Lyapunov exponents’ criteria to develop chaos diagram on the parameters space of 4-dimensional harmonically excited vibration absorber control Duffing’s Oscillator. Relevant simulations were effected by choice combination of constant step Runge-Kutta methods and Grahm Schmidt Orthogonal rules. Simulations of 4-dimensional hyper-chaotic models of modified Lorenz and ...

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...

Additive Runge Kutta (ARK) methods are investigated for application to the spatially discretized one dimensional convection diffusion reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge Kutta methods are grouped into a single composite method. Then, implicit explicit, N = 2, additive Runge Kutta (ARK2) m...

Presenting and simulating the numerical treatment of nine-dimensional fractional chaotic Lorenz system is goal this work. The spectral collocation method (SCM), which makes use Changhee polynomials Appell type, suggested approximation technique to achieve goal. A rough formula for Caputo derivative first derived, it used build strategy model’s solution. This procedure creates a algebraic equati...

Based on reasonable testing model problems, we study the preservation by symplectic Runge-Kutta method (SRK) and symplectic partitioned Runge-Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The structure-preservation region provides a practical criterion for choosing step-size in symplectic computation. Examples are given to justify the investigation.

This paper presents a family of Runge{Kutta type integration schemes of arbitrarily high order for di erential equations evolving on manifolds. We prove that any classical Runge{Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.

In this study, we construct new numerical methods for solving the initial value problem (IVP) in ordinary differential equations based on a symmetrical quadrature integration formula using hybrid functions. The proposed are designed to provide an efficient and accurate solution IVP more suitable problems with non-smooth solutions. key idea behind is combine advantages of traditional methods, su...

In this paper we study a class of explicit pseudo two-step Runge-Kutta (EP-TRK) methods for rst-order ODEs for parallel computers. We investigate linear stability and derive methods with enlarged stability regions. In numerical experiments on a shared memory computer we compare a parallel variable step size EPTRK implementation with the eecient sequential Runge-Kutta method dopri5.