Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we a...

In this article we proposed three explicit Improved Runge-Kutta (IRK) methods for solving first-order ordinary differential equations. These methods are two-step in nature and require lower number of stages compared to the classical RungeKutta method. Therefore the new scheme is computationally more efficient at achieving the same order of local accuracy. The order conditions of the new methods...

The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obta...

In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the θ-methods with 1 2 ≤ θ ≤ 1, the odd stage Gauss-Legendre m...

We study the stability of Runge-Kutta methods for the time integration of semidiscrete systems associated with time dependent PDEs. These semidiscrete systems amount to large systems of ODEs with the possibility that the matrices involved are far from being normal. The stability question of their Runge-Kutta methods, therefore, cannot be addressed by the familiar scalar arguments of eigenvalues...