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...

In this paper, we employed both first and second laws of thermodynamics to analyze the flow and thermal decomposition in a variable viscosity Couette flow of a conducting fluid in a rotating system under the combined influence of magnetic field and Hall current. The non-linear governing differential equations are obtained and solved numerically using shooting method coupled with fourth order Ru...

in this paper, we investigate the total variation diminishing property for a class of 2-stage explicit rung-kutta methods of order two (rk2) when applied to the numerical solution of special nonlinear initial value problems (ivps) for (odes). schemes preserving the essential physical property of diminishing total variation are of great importance in practice. such schemes are free of spurious o...

In this paper, we investigate the total variation diminishing property for a class of 2-stage explicit Rung-Kutta methods of order two (RK2) when applied to the numerical solution of special nonlinear initial value problems (IVPs) for (ODEs). Schemes preserving the essential physical property of diminishing total variation are of great importance in practice. Such schemes are free of spurious o...

Sugiura, H. and T. Torii, A method for constructing generalized Runge-Kutta methods, Journal of Computational and Applied Mathematics 38 (1991) 399-410. In the implementation of an implicit Runge-Kutta formula, we need to solve systems of nonlinear equations. In this paper, we analyze the Newton iteration process and a modified Newton iteration process for solving these equations. Then we propo...

A set of validated numerical integration methods based on explicit and implicit Runge-Kutta schemes is presented to solve, in a guaranteed way, initial value problems of ordinary differential equations. Runge-Kutta methods are well-known to have strong stability properties, which make them appealing to be the basis of validated numerical integration methods. A new approach to bound the local tr...

Exponential Runge-Kutta (ERK) and partitioned exponential Runge-Kutta (PERK) 4 methods are developed for solving initial value problems with vector fields that can be split into con5 servative and linear non-conservative parts. The focus is on linearly damped ordinary differential 6 equations, that possess certain invariants when the damping coefficient is zero, but, in the presence of 7 consta...

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...

Abstract— General linear methods (GLM) apply to a large family of numerical methods for ordinary differential equations, with RungeKutta (RK) and Almost Runge-Kutta (ARK) methods as two out of a number of special cases. In this paper, we have investigated the efficacy of Richardson extrapolation (RE) technique as a means of obtaining viable and acceptable estimates of the local truncation error...