-A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step. ...

The construction of new explicit Runge–Kutta methods taking into account, not only their accuracy, but also the preservation of Quadratic Invariants (QIs) is studied. An expression of the error of conservation of a QI by a Runge–Kutta method is given, and a new six–stage formula with classical order four and seventh order of QI–conservation is obtained by choosing their coefficients so that the...

The e ciency and accuracy of several time integration schemes are investigated for the unsteady Navier-Stokes equations. This study focuses on the e ciency of higher-order Runge-Kutta schemes in comparison with the popular Backward Di erencing Formulations. For this comparison an unsteady two-dimensional laminar ow problem is chosen, i.e. ow around a circular cylinder at Re=1200. It is conclude...

We study semilinear evolution equations [Formula: see text] posed on a Hilbert space [Formula: see text], where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from [Formula: see text] to itself, and [Formula: see text], [Formula: see text], [Formula: see text]. In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation wi...

The use of computer algebra systems in a course on scientific computation is demonstrated. Various examples, such as the derivation of Newton’s iteration formula, the secant method, Newton–Cotes and Gaussian integration formulas, as well as Runge–Kutta formulas, are presented. For the derivations, the computer algebra system Maple is used.

This paper focuses on the mathematical study of the existence of solitary gravity waves (solitons) and their characteristics (amplitude, velocity, [Formula: see text]) generated by a piston wave maker lying upstream of a horizontal channel. The mathematical model requires both incompressibility condition, irrotational flow of no viscous fluid and Lagrange coordinates. By using both the inverse ...

The multiplicative structure of the trivial symplectic groupoid over R associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable condi...

We examine the potential for parallelism in Runge-Kutta (RK) methods based on formulas in standard one-step form. Both negative and positive results are presented. Many of the negative results are based on a theorem that bounds the order of a RK formula in terms of the minimum polynomial for its coeecient matrix. The positive results are largely examples of prototypical formulas which ooer a po...

This paper deals with a special family of implicit Runge–Kutta formulas of orders 2, 4 and 6. These methods are of Gauss type; i.e., they are based on Gauss quadrature formulas of orders 2, 4 and 6, respectively. However, the methods under discussion have only explicit internal stages that lead to cheap practical implementation. Some of the stage values calculated in a step of the numerical int...

Among the most popular methods for the solution of the Initial Value Problem are the Runge–Kutta pairs of orders 5 and 4. These methods can be derived solving a system of nonlinear equations for its coefficients. For achieving this, we usually admit various simplifying assumptions. The most common of them are the so called row simplifying assumptions. Here we negligible them and present an algo...