INTEGRATION FOR SPECIAL THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS USING IMPROVED RUNGE-KUTTA DIRECT METHOD

Fifth Order Improved Runge-Kutta Method for Solving Ordinary Differential Equations

Abstract: In this paper, the fifth order Improved Runge-Kutta method (IRK5) that uses just five function evaluations per step is developed. The method proposed here are derived with only five stages which results in lower number of function evaluations. Therefore, IRK5 has a lower computational cost than the classical fifth order Runge-Kutta method (RK5). Here, the order conditions of the metho...

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...

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

Additive Runge-Kutta Methods for Stiff Ordinary Differential Equations

Certain pairs of Runge-Kutta methods may be used additively to solve a system of n differential equations x' = J(t)x + g(t, x). Pairs of methods, of order p < 4, where one method is semiexplicit and /(-stable and the other method is explicit, are obtained. These methods require the LU factorization of one n X n matrix, and p evaluations of g, in each step. It is shown that such methods have a s...

Implicit Runge-Kutta Methods for Lipschitz Continuous Ordinary Differential Equations

Implicit Runge-Kutta(IRK) methods for solving the nonsmooth ordinary differential equation (ODE) involve a system of nonsmooth equations. We show superlinear convergence of the slanting Newton method for solving the system of nonsmooth equations. We prove the slanting differentiability and give a slanting function for the involved function. We develop a new code based on the slanting Newton met...