Numerical solutions of the generalized Burgers-Fisher equation are presented based on a polynomial-based differential quadrature method with minimal computational effort. To achieve this, a combination of a polynomial-based differential quadrature method in space and a third-order strong stability preserving Runge-Kutta scheme in time have been used. The proposed technique successfully worked t...

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial-boundary value problems with time-dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order of convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved and computati...

This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge — Kutta method (described in [12]) for treating the differential part and the collocation method (using Boole’s quadrature rule) for treating the integral part. The efficie...

This paper continues earlier work by the same author concerning the stability and B-convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, Bor optimally B-convergent ...

In this paper we consider the numerical solution of taxis–diffusion–reaction models. Such nonlinear partial differential equation models appear often in mathematical biology and we consider examples from tumour growth and invasion, and the aggregation of amoebae. These examples are characterised by a taxis term which is large in magnitude compared with the diffusion term and this can cause nume...

Users of locally-adaptive software for initial value ordinary differential equations are likely to be concerned with global errors. At the cost of extra computation, global error estimation is possible. Zadunaisky's method and 'solving for the error estimate' are two techniques that have been successfully incorporated into Runge-Kutta algorithms. The standard error analysis for these techniques...

In this paper we present the analysis for the Runge-Kutta discontinuous Galerkin (RKDG) method to solve scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge–Kutta (TVDRK3) method. We use an energy technique to present the L-norm stability for scalar linear conservation laws, and obtain a priori error estimates for smooth solutions...

Received: December 9, 2010 Accepted: January 6, 2011 doi:10.5539/jmr.v3n2p126 Abstract Stepwise local error control using local extrapolation in Runge-Kutta methods is well-known. In this paper, we introduce an algorithm, designated RKrvQz, that is capable of controlling local and global errors in a stepwise manner. The algorithm utilizes three Runge-Kutta methods, of orders r, v and z, with r ...

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...

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...