Abstract

A major objective of the study described in the present paper is to achieve a high accuracy in the time integration of transient convection-diffusion problems and to eventually combine this with new methods for space discretization, such as meshless methods. In this way, a uniformly high-order accurate methodology could be made available for the numerical solution of convection-diffusion problems. Both Padé approximations of the exponential function and Runge-Kutta methods are considered for deriving multi-stage schemes involving first time derivatives only, thus easier to implement than standard Taylor-Galerkin schemes which incorporate second and third time derivatives. After a brief discussion of the stability and accuracy properties of the stability and accuracy properties of the multi-stage schemes and the presentation of illustrative examples, the paper closes with some considerations on the coupling of high-order accurate temporal schemes and meshless methods for the spatial representation.  

Abstract

The paper discusses the formulation of high-order accurate time-stepping schemes for transient convection–diffusion problems to be combined with finite element methods of the least-squares type for a stable discretization of highly convective problems. Padé approximations of the exponential function are considered for deriving multi-stage time integration schemes involving first time derivatives only, thus easier to implement in conjunction with C0 finite elements than standard time-stepping schemes which incorporate higher-order time derivatives. After a brief discussion of the stability and accuracy properties of the multi-stage Padé schemes and having underlined the similarity between Padé and Runge–Kutta methods, the paper closes with the presentation of illustrative examples which indicate the effectiveness of the proposed methods.

Abstract

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and WrightΩ functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.