Numerical study of third-order ordinary differential equations using a new class of two derivative Runge-Kutta type methods

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...

Runge-Kutta Methods for Linear Ordinary Differential Equations

Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODEs) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficient...

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

Block Runge-Kutta Methods for the Numerical Integration of Initial Value Problems in Ordinary Differential Equations

Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of v...