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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 purpose of this work is to investigate the similarity requirements for the application of multifidelity modeling (MFM) for the prediction of airfoil dynamic stall using computational fluid dynamics (CFD) simulations.

Dynamic stall is modeled using the unsteady Reynolds-averaged Navier–Stokes equations and Menter's shear stress transport turbulence model. Multifidelity models are created by varying the spatial and temporal discretizations. The effectiveness of the MFM method depends on the similarity between the high- (HF) and low-fidelity (LF) models. Their similarity is tested by computing the prediction error with respect to the HF model evaluations. The proposed approach is demonstrated on three airfoil shapes under deep dynamic stall at a Mach number 0.1 and Reynolds number 135,000.

The results show that varying the trust-region (TR) radius (λ) significantly affects the prediction accuracy of the MFM. The HF and LF simulation models hold similarity within small (λ ≤ 0.12) to medium (0.12 ≤ λ ≤ 0.23) TR radii producing a prediction error less than 5%, whereas for large TR radii (0.23 ≤ λ ≤ 0.41), the similarity is strongly affected by the time discretization and minimally by the spatial discretization.

The findings of this work present new knowledge for the construction of accurate MFMs for dynamic stall performance prediction using LF model spatial- and temporal discretization setup and the TR radius size. The approach used in this work is general and can be used for other unsteady applications involving CFD-based MFM and optimization.

The purpose of this work is to present a procedure for determining the wind drift factor through two-dimensional computational fluid dynamics (CFD) simulations of the wind acting on a wavy sea surface, such that the subjectivity of its estimation is reduced.

The wind drift factor was determined by two-dimensional CFD analyses with open-channel condition. The characteristic wave was determined by the Sverdrup–Munk–Bretschneider (SMB) method. The uncertainty analysis is based on convergence studies using a single parameter refinement (grid and time step).

This procedure allows the estimation of the wind drift factor in a fetch-limited domain. The domain's value in the analyzed region is 0.0519 ± 4.92% which is consistent with the upper values of the wind drift factors reported in the literature.

The use of a three-dimensional domain was impractical with the available computational resources because of the fine mesh required for wave modeling. The uncertainty analysis consisted only of a verification procedure. Validation against real data was not possible because of the lack of measured data in the analyzed region.

The wind drift factor is usually estimated based on either experience or random sampling. The original contribution of this work is the presentation of a CFD procedure for estimating the wind drift factor, in which the domain inlet is subjected to a wave boundary condition and to a wind velocity.

This study aims to investigate the insights of soot formation such as rate of soot coagulation, rate of soot nucleation, rate of soot surface growth and soot surface oxidation in ethylene/hydrogen/nitrogen diffusion jet flame at standard atmospheric conditions, which is very challenging to capture even with highly sophisticated measuring systems such as Laser Induced Incandescence and Planar laser-induced fluorescence. The study also aims to investigate the volume of soot in the flame using soot volume fraction and to understand the global correlation effect in the formation of soot in ethylene/hydrogen/nitrogen diffusion jet flame.

A large eddy simulation (LES) was performed using box filtered subgrid-scale tensor. A filtered and residual component of the governing equations such as continuity, momentum, energy and species are resolved and modeled, respectively. All the filtered and residual components are numerically solved using the ILU method by considering PISO pressure–velocity solver. All the hyperbolic flux uses the QUICK algorithm, and an elliptic flux uses SOU to evaluate face values. In all the cases, Courant–Friedrichs–Lewy (CFL) conditions are maintained unity.

The findings are as follows: soot volume fraction (SVF) as a function of a flame-normalized length for three different Reynolds number configurations (Re = 15,000, Re = 8,000 and Re = 5,000) using LES; soot gas phase and particulate phase insights such as rate of soot nucleation, rate of soot coagulation, rate of soot surface growth and soot surface oxidation for three different Reynolds number configurations (Re = 15,000, Re = 8,000 and Re = 5,000); and soot global correction using total soot volume in the flame volume as a function of Reynolds number and Froude number.

The originality of this study includes the following: coupling LES turbulent model with chemical equilibrium diffusion combustion conjunction with semi-empirical Brookes Moss Hall (BMH) soot model by choosing C6H6 as a soot precursor kinetic pathway; insights of soot formations such as rate of soot nucleation, soot coagulation rate, soot surface growth rate and soot oxidation rate for ethylene/hydrogen/nitrogen co-flow flame; and SVF and its insights study for three inlet fuel port configurations having the three different Reynolds number (Re = 15,000, Re = 8,000 and Re = 5,000).

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.

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.

Semi-implicit type cutting plane method (CPM) and fully implicit type closest point projection method (CPPM) are the two most widely used frameworks for numerical stress integration. CPM is simple, easy to implement and accurate up to first order. CPPM is unconditionally stable and accurate up to second order though the formulation is complex. Therefore, this study aims to develop a less complex and accurate stress integration method for complex constitutive models.

Two integration techniques are formulated using the midpoint and Romberg method by modifying CPM. The algorithms are implemented for three different classes of soil constitutive model. The efficiency of the algorithms is judged via stress point analysis and solving a boundary value problem.

Stress point analysis indicates that the proposed algorithms are stable even with a large step size. In addition, numerical analysis for solving boundary value problem demonstrates a significant reduction in central processing unit (CPU) time with the use of the semi-implicit-type midpoint algorithm.

Traditionally, midpoint and Romberg algorithms are formulated from explicit integration techniques, whereas the present study uses a semi-implicit approach to enhance stability. In addition, the proposed stress integration algorithms provide an efficient means to solve boundary value problems pertaining to geotechnical engineering.

This paper aims to point out the critical problems in numerical verification of solidification simulation codes and the complexity of the verification and to propose and apply a procedure of generalized verification for macrosegregation simulation.

A partial verification of a finite‐volume computational model of macrosegregation in direct chill (DC) casting of binary aluminum alloys, including the coupled transport phenomena of heat transfer, fluid flow and species transport, is performed. The verification procedure is conducted on numerical test problems, defined as subproblems with respect to the complexity of the physical model, geometry, and boundary conditions. The studied cases are thermal convection with solidification in DC casting, thermal natural convection of a low‐Prandtl‐number liquid metal in a rectangular cavity and 1D directional solidification of a binary Al‐Cu alloy. Grid‐convergence studies, code comparison with an alternative Chebyshev‐collocation method, and comparison with a reference similarity solution are used for verification.

An excellent ability of the model to accurately resolve the thermal convection in the pertinent range of Prandtl and Rayleigh numbers is shown. Concerns regarding the solution of species transport in the mushy zone remain.

The proposed verification procedure is not completed in its entirety. Further verification of the solutal and thermosolutal convection problems is required.

This paper proposes verification techniques for complex coupled solidification problems involving significant convection in the melt.

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.

This paper presents a new approach for estimating the discretization error of finite element analysis of generalized eigenproblems. The method uses smoothed gradients at nodal points to derive improved element‐by‐element interpolation functions. The improved interpolation functions and their gradients are used in the Rayleigh quotient to obtain an improved eigenvalue. The improved eigenvalue is used to estimate the error of the original solution. The proposed method does not require any re‐solution of the eigenproblem. Results for 1‐D and 2‐D C° eigenproblems in acoustics and elastic vibrations are used as examples to demonstrate the accuracy of the proposed method.