We introduce new projective versions of second-order accurate Runge-Kutta and Adams-Bashforth methods, and demonstrate their use as outer integrators in solving stiff differential systems. An important outcome is that the new outer integrators, when combined with an inner telescopic projective integrator, can result in fully explicit methods with adaptive outer step size selection and solution ...

This paper introduces a three and a four order explicit time stepping method. These methods have high stage order and favorable monotonicity properties. The proposed methods are based on multistagemultistep (MM) schemes that belong to the broader class of general linear methods, which are generalizations of both Runge-Kutta and linear multistep methods. Methods with high stage order alleviate t...

Implicit-explicit (IMEX) multistep methods are very useful for the time discretiza-tion of convection diffusion PDE problems such as the Burgers equations and also the incompressible Navier-Stokes equations. Semi-discretization in space of the latter typically gives rise to an index 2 differential-algebraic (DAE) system of equations. Runge-Kutta (RK) methods have been considered for the time di...

Last time, we investigated the fourth-order Runge-Kutta method. We saw that the computations involved in performing this approximation were less than ideal. To create more computationally viable methods, we introduced multistep methods, in which the approximation at a given point is obtained using only the values of the differential equation to be approximated and the approximation itself at pr...

In this article, there is an attempt to introduce, generally, spectral methods for numerical solution of ordinary differential equations, also, to focus on those problems in which some coefficient functions or solution function is not analytic. Then, by expressing weak and strong aspects of spectral methods to solve this kind of problems, a modified spectral method which is more efficient than ...

Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving partial differential equations (PDEs) with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily impleme...

The Taguchi method of product design is an experimental approximation to minimizing the expected value of target variance for certain classes of problems. Taguchi’s method is extended to designs which involve variables each of which has a range of values all of which must be satisfied (necessity), and designs which involve variables each of which has a range of values any of which might be used...

In this paper, we describe stochastic Runge–Kutta (SRK) methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations (SDEs) which was first introduced by Burrage and Burrage in 1996. In particular, three new SRK methods with strong order 1.0 are constructed. They are an explicit two–stage method, an explicit three–stage method with minimum principal error...

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...

A one-step 4-stage Hermite-Birkhoff-Taylor method of order 12, denoted by HBT(12)4, is constructed for solving nonstiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. The method uses derivatives y′ to y(9) as in Taylor methods combined with a 4-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the tr...