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The purpose of this paper is to present outcomes of efforts made over the last 20 years to extend the applicability of the Richardson extrapolation procedure to numerical predictions of multidimensional, steady and unsteady, fluid flow and heat transfer phenomena in regular and irregular calculation domains.

Pattern-preserving grid-refinement strategies are proposed for mathematically rigorous generalizations of the Richardson extrapolation procedure for numerical predictions of steady fluid flow and heat transfer, using finite volume methods and structured multidimensional Cartesian grids; and control-volume finite element methods and unstructured two-dimensional planar grids, consisting of three-node triangular elements. Mathematically sound extrapolation procedures are also proposed for numerical solutions of unsteady and boundary-layer-type problems. The applicability of such procedures to numerical solutions of problems with curved boundaries and internal interfaces, and also those based on unstructured grids of general quadrilateral, tetrahedral, or hexahedral elements, is discussed.

Applications to three demonstration problems, with discretizations in the asymptotic regime, showed the following: the apparent orders of accuracy were the same as those of the numerical methods used; and the extrapolated results, measures of error, and a grid convergence index, could be obtained in a smooth and non-oscillatory manner.

Strict or approximate pattern-preserving grid-refinement strategies are used to propose generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. Such extrapolation procedures play an indispensable role in the verification and validation techniques that are employed to assess the accuracy of numerical predictions which are used for designing, optimizing, virtual prototyping, and certification of thermofluid systems.

The potential for overall efficiency improvements of modern hydro power turbines is a few percent. A significant part of the losses occurs in the draft tube. To improve the efficiency by analysing the flow in the draft tube, it is therefore necessary to do this accurately, i.e. one must know how large the iterative and the grid errors are. This was done by comparing three different methods to estimate errors. Four grids (122,976 to 4,592 cells) and two numerical schemes (hybrid differencing and CCCT) were used in the comparison. To assess the iterative error, the convergence history and the final value of the residuals were used. The grid error estimates were based on Richardson extrapolation and least square curve fitting. Using these methods we could, apart from estimate the error, also calculate the apparent order of the numerical schemes. The effects of using double or single precision and changing the under relaxation factors were also investigated. To check the grid error the pressure recovery factor was used. The iterative error based on the pressure recovery factor was very small for all grids (of the order 10^–4 percent for the CCCT scheme and 10^–10percent for the hybrid scheme). The grid error was about 10 percent for the finest grid and the apparent order of the numerical schemes were 1.6 for CCCT (formally second order) and 1.4 for hybrid differencing (formally first order). The conclusion is that there are several methods available that can be used in practical simulations to estimate numerical errors and that in this particular case, the errors were too large. The methods for estimating the errors also allowed us to compute the necessary grid size for a target value of the grid error. For a target value of 1 percent, the necessary grid size for this case was computed to 2 million cells.

Extrapolation is a process used to accelerate the convergence of a sequence of approximations to the true value. Different stepsizes are used to obtain approximate solutions, which are combined to increase the order of the approximation by eliminating leading error terms. The smoothing technique is also applied to suppress order reduction and to dampen the oscillatory component in the numerical solution when solving stiff problems. The extrapolation and smoothing technique can be applied in either active, passive or the combination of both active and passive modes. In this paper, the authors investigate the best strategy of implementing extrapolation and smoothing technique and use this strategy to solve stiff ordinary differential equations. Based on the experiment, the authors suggest using passive smoothing in order to reduce the computation time.

The two-step smoothing is a composition of four steps of the symmetric method with different weights. It is used as the final two steps when combined with many steps of the symmetric method. The aim is to preserve symmetry and provide damping for stiff problem and to be more robust than the one-step smoothing. The two-step smoothing is L-stable. The new method is then applied with extrapolation process in passive and active modes to investigate the most efficient and accurate method of implementation.

In this paper, the authors constructed the two-step smoothing to be more robust than the one-step smoothing. The two-step smoothing is constructed to achieve as high order as possible and able to restore the classical order of particular method compared to the one-step active smoothing that is only able to achieve order-1 condition. The two-step smoothing for ITR is also superior in solving stiff case since it has the super-convergent order-4 behavior. In our experiments with extrapolation, it is proven that the two-step smoothing is more accurate and more efficient than the one-step smoothing, namely 1ASAX. It is also observed that the method with smoothing is comparable if not superior to the existing base method in certain cases. Based on the experiment, the authors would suggest using passive smoothing if the aim is to reduce computation time. It is of interest to conduct more experiment to validate the accuracy and efficiency of the smoothing formula with and without extrapolation.

The implementation of extrapolation on two-step symmetric Runge–Kutta method has not been tested on variety of other test problems yet. The two-step symmetrization is an extension of the one-step symmetrization and has not been constructed by other researchers yet. The method is constructed such that it preserves the asymptotic error expansion in even powers of stepsize, and when used with extrapolation the order might increase by 2 at a time. The method is also L-stable and eliminates the order reduction phenomenon when solving stiff ODEs. It is also of interest to observe other ways of implementing extrapolation using other sequences or with interpolation.

The purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point.

The authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme.

The proposed method has a convergence rate of order O(N−2L2+Δt). Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor.

A class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.

The paper aims to propose several new approaches for the discrete‐time integration of stiff non‐linear differential equations.

The proposed approaches build on a method developed by the authors involving Padé approximates about each function sample. Both single‐ and multi‐step methods are suggested. The use of Richardson extrapolation is recommended for increasing efficiency.

The efficacy of the methods is shown using two examples and results are compared to a standard integration technique.

The paper shows that the methods are suitable for application in any field of science requiring efficient and accurate numerical solution of stiff differential equations.

We present an approach for pricing American put options with a regime-switching volatility. Our method reveals that the option price can be expressed as the sum of two components: the price of a European put option and the premium associated with the early exercise privilege. Our analysis demonstrates that, under these conditions, the perpetual put option consistently commands a higher price during periods of high volatility compared to those of low volatility. Moreover, we establish that the optimal exercise boundary is lower in high-volatility regimes than in low-volatility regimes. Additionally, we develop an analytical framework to describe American puts with an Erlang-distributed random-time horizon, which allows us to propose a numerical technique for approximating the value of American puts with finite expiry. We also show that a combined approach involving randomization and Richardson extrapolation can be a robust numerical algorithm for estimating American put prices with finite expiry.

Live non-invasive monitoring of biomarkers is of great importance for the medical community. Moreover, some studies suggest that there is a substantial business gap in the development of mass-production commercial sweat-analysing wearables with great revenue potential. The objective of this work is to quantify the concentration of biomarkers that reaches the area of the garment where a sensor is positioned to advance the development of commercial sweat-analysing garments.

Computational analysis of the microfluidic transport of biomarkers within eccrine sweat glands provides a powerful way to explore the potential for quantitative measurements of biomarkers that can be related to the health and/or the physical activity parameters of an individual. The numerical modelling of sweat glands and the interaction of sweat with a textile layer remain however rather unexplored. This work presents a simulation of the production of sweat in the eccrine gland, reabsorption from the dermal duct into the surrounding skin and diffusion within an overlying garment.

The model represents satisfactorily the relationship between the biomarker concentration and the flow rate of sweat. The biomarker distribution across an overlying garment has also been calculated and subsequently compared to the minimum amount detectable by a sensor previously reported in the literature. The model can thus be utilized to check whether or not a given sensor can detect the minimum biomarker concentration threshold accumulated on a particular type of garment.

The present work presents to the best of our knowledge, the earliest numerical models of the sweat gland carried out so far. The model describes the flow of human sweat along the sweat duct and on to an overlying piece of garment. The model considers complex phenomena, such as reabsorption of sweat into the skin layers surrounding the duct, and the structure of the fibres composing the garment. Biomarker concentration maps are obtained to check whether sensors can detect the threshold concentration that triggers an electric signal. This model finds application in the development of smart textiles.

The purpose of this paper is to discuss the need to attend correctly to the accuracy and the manner in which the value of the streamfunction is determined when two or more impermeable boundaries are present. This is discussed within the context of the paper by Nandalur et al. (2019), which concerns the effect of a centrally located conducting square block on convection in a square sidewall-heated porous cavity. Detailed solutions are also presented which allow the streamfunction to take the natural value on the surface of the internal block.

Steady solutions are obtained using finite difference methods. Three different ways in which insulating boundary conditions are implemented are compared. Detailed attention is paid to the iterative convergence of the numerical scheme and to its overall accuracy. Error testing and Richardson’s extrapolation have been used to obtain very precise values of the Nusselt number.

The assumption that the streamfunction takes a zero value on the boundaries of both the cavity and the embedded block is shown to be incorrect. Application of the continuity-of-pressure requirement shows that the block and the outer boundary take different constant values.

The Darcy–Rayleigh number is restricted to values at or below 200; larger values require a finer grid.

This paper serves as a warning that one cannot assume that the streamfunction will always take a zero value on all impermeable surfaces when two or more are present. A systematic approach to accuracy is described and recommended.

The purpose of this paper is to present a numerical and mathematical model of a moulding process of a dry electrical transformer. Moreover, the calculated results are reported and compared with experimental measurements.

An experimental rig, for carrying out and monitoring a moulding process, has been designed and built. Two experiments were preformed. First was an isothermal experiment in which an analog liquid was used. The second experiment was a non‐isothermal one in which an epoxy resin was used. For the rig geometry, the numerical mesh, with the use of the commercial code Gambit, was built. All necessary physical properties, including viscosity, surface tension and contact angle of fluids used in the experiments were measured.

The Euler approach for modelling multiphase flow with a free surface is addressed in the presented work. Comparison of the computational results with measurements on the designed experimental rig revealed good agreement. Comparison was carried out through measurements of free surface characteristic features captured with a digital camera and through temperature measurements for the nonisothermal case. Richardson extrapolation method was successfully applied to estimate the numerical discretisation error, proving that a grid independent solution was obtained.

This paper is useful for researchers and industrialists involved in the modelling of moulding processes, giving guidance on the available mathematical models appropriate for this kind of problem. Moreover, it provides valuable information as to how to perform validation and verification procedures for such real‐life processes.

This paper aims to solve inhomogeneous dielectric problems by matching boundary conditions at the interfaces among homogeneous subdomains. The capabilities of Hilbert transform computations are deeply investigated in the case of limited numbers of samples, and a refined model is presented by means of investigating accuracies in a case study with three subdomains.

The accuracies, refined by Richardson extrapolation to zero error, are compared to finite element (FEM) and finite difference methods. The boundary matching procedures can be easily applied to the results of a previous Schwarz–Christoffel (SC) conformal mapping stage in SC + BC procedures, to cope with field singularities or with open boundary problems.

The proposed field computations are of general interest both for electrostatic and magnetostatic field analysis and optimization. They can be useful as comparison tools for FEM results or when severe field singularities can impair the accuracies of other methods.

This static field methodology, of course, can be used to analyse transverse electro magnetic (TEM) or quasi-TEM propagation modes. It is possible that, in some case, these may make a contribution to the analysis of axis symmetrical problems.

The most relevant result is the possible introduction of SC + BC computations as a standard tool for solving inhomogeneous dielectric field problems.