Embedded explicit Runge-Kutta type methods for directly solving special third order differential equations y'''=f(x, y)

Embedded explicit Runge – Kutta type methods for directly solving special third order differential equations

A survey of pseudo Runge-Kutta methods

This survey collects the theoretical results in the area of pseudo RungeKutta methods (PRK ) for ordinary differential equations and it is a vehicle for a current bibliography from 1966 to 2002. PRK methods require fewer functional evaluations than Runge-Kutta methods of the same order. Byrne and Lambert (1966-1967 ) was the first who considered PRK methods in significative forms. Afterwards Co...

Nonstandard explicit third-order Runge-Kutta method with positivity property

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) pos...

2-stage explicit total variation diminishing preserving Runge-Kutta methods

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...

On Those Ordinary Differential Equations That Are Solved Exactly by the Improved Euler Method

As a numerical method for solving ordinary differential equations y′ = f(x, y), the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for f(x, y) to a finite system that is sufficient and necessary for the improved ...