A parallel DIRK method for stiff initial-value problems

AbstractIn this note we propose a fast parallel iteration process for solving a low-order implicit Runge–Kutta method. The resulting scheme can be regarded as a parallel singly diagonally implicit Runge–Kutta (PDIRK) method. On a two-processor computer, this method requires effectively the solution of two implicit relations per step. By two numerical experiments we compare this method with some sequential methods from the literature, and show its efficient behaviour

Continuous variable stepsize explicit pseudo two-step RK methods

AbstractThe aim of this paper is to apply a class of constant stepsize explicit pseudo two-step Runge-Kutta methods of arbitrarily high order to nonstiff problems for systems of first-order differential equations with variable stepsize strategy. Embedded formulas are provided for giving a cheap error estimate used in stepsize control. Continuous approximation formulas are also considered for use in an eventual implementation of the methods with dense output. By a few widely used test problems, we compare the efficiency of two pseudo two-step Runge-Kutta methods of orders 5 and 8 with the codes DOPRI5, DOP853 and PIRK8. This comparison shows that in terms of ƒ-evaluations on a parallel computer, these two pseudo two-step Runge-Kutta methods are a factor ranging from 3 to 8 cheaper than DOPRI5, DOP853 and PIRK8. Even in a sequential implementation mode, fifth-order new method beats DOPRI5 by a factor more than 1.5 with stringent error tolerances

RKN-type parallel block PC methods with Lagrange-type predictors

AbstractThis paper describes the construction of block predictor-corrector methods based on Runge-Kutta-Nyström correctors. Our approach is to apply the predictor-corrector method not only at step points, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-order predictor formula using Lagrange interpolation. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictor formulas based on last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta-Nyström corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor-corrector methods to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature

A general class of explicit pseudo two-step RKN methods on parallel computers

AbstractThe aim of this paper is to investigate a general class of explicit pseudo two-step Runge-Kutta-Nyström methods (RKN methods) of arbitrarily high order for nonstiff problems for systems of special second-order differential equations y″(t) = f(y(t)). Order and stability considerations show that we can obtain for any given p, a stable pth-order explicit pseudo two-step RKN method requiring p − 2 right-hand side evaluations per step of which each evaluation can be obtained in parallel. Consequently, on a multiprocessor computer, only one sequential right-hand side evaluation per step is required. By a few widely-used test problems, we show the superiority of the methods considered in this paper over both sequential and parallel methods available in the literature

CONNECTING MATHEMATICS AND PRACTICE: A CASE STUDY OF TEACHING EXPONENTIAL FUNCTIONS

There is a need for teaching exponential functions to show the necessity for a better match between the knowledge of exponential functions in high schools with the practical application of it in fields. In this research, a teaching process was built in association with teaching situations to show students the relationship between mathematics and real life. The research sample included 76 students in high schools in Vietnam. Additionally, two problems of compound interest and population growth were integrated and were the main research instruments. Data were collected, including student work, and they were analyzed qualitatively. The results showed that students had improved their problem-solving skills and saw the relationship between mathematics and practice. Furthermore, there were some recommendations suggested for textbook authors and teachers. Article visualizations

AMMONIA REMOVAL FROM SWINE WASTEWATER USING AN AEROBIC, ANOXIC FILTER AT A PILOT-SCALE IN THANH LOC BIOSTATION

Joint Research on Environmental Science and Technology for the Eart