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This paper aims to focus on the temporal and spatial fourth‐order finite volume discretization of the incompressible form of the Navier‐Stokes equations on structured uniform grids. The main purpose of the paper is to assess the accuracy enhancement with the inclusion of a high‐order reconstruction of the point‐wise velocity field on a fourth‐order accurate numerical scheme for the solution of the unsteady incompressible Navier‐Stokes equations.

The present finite volume method uses a fractional time‐step for decoupling velocity and pressure. A Runge‐Kutta integration scheme is implemented for integrating the momentum equation along with a polynomial interpolation and Simpson formula for space‐integration. The formulation is based on step‐by‐step de‐averaging process applied to the velocity field.

The reconstruction of the point‐wise velocity field on a higher‐order basis is essential to obtain solutions that effectively stand for a fourth‐order approximation of the point‐wise one. Results are provided for the Taylor vortex decay problem and for co‐ and counter‐rotating vortices to assess the increase in accuracy promoted by the inclusion of the high‐order de‐averaging procedure.

High‐order reconstruction of the point‐wise velocity field should be considered in high‐order finite volume methods for the solution of the unsteady incompressible form of the Navier‐Stokes equations on structured grids.

The inclusion of a high‐order reconstruction of the point‐wise velocity field is a simple and effective method of enhancing the accuracy of a finite volume code for the computational fluid dynamics analysis.

The paper develops an improved version of a fourth‐order accurate finite volume projection method with the inclusion of a high‐order reconstruction step.

This paper aims to study the sampling methods (or design of experiments) which have a large influence on the performance of the surrogate model. To improve the adaptability of modelling, a new sequential sampling method termed as sequential Chebyshev sampling method (SCSM) is proposed in this study.

The high-order polynomials are used to construct the global surrogated model, which retains the advantages of the traditional low-order polynomial models while overcoming their disadvantage in accuracy. First, the zeros of Chebyshev polynomials with the highest allowable order will be used as sampling candidates to improve the stability and accuracy of the high-order polynomial model. In the second step, some initial sampling points will be selected from the candidates by using a coordinate alternation algorithm, which keeps the initial sampling set uniformly distributed. Third, a fast sequential sampling scheme based on the space-filling principle is developed to collect more samples from the candidates, and the order of polynomial model is also updated in this procedure. The final surrogate model will be determined as the polynomial that has the largest adjusted R-square after the sequential sampling is terminated.

The SCSM has better performance in efficiency, accuracy and stability compared with several popular sequential sampling methods, e.g. LOLA-Voronoi algorithm and global Monte Carlo method from the SED toolbox, and the Halton sequence.

The SCSM has good performance in building the high-order surrogate model, including the high stability and accuracy, which may save a large amount of cost in solving complicated engineering design or optimisation problems.

This paper aims to provide a simple and accurate higher order predictor‐corrector integration which can be used in dynamic analysis and to compare it with the previous works.

The predictor‐corrector integration is defined by combining the higher order explicit and implicit integrations in which displacement and velocity are assumed to be functions of accelerations of several previous time steps. By studying the accuracy and stability conditions, the weighted factors and acceptable time step are determined.

Simplicity and vector operations plus accuracy and stability are the main specifications of the new predictor‐corrector method. This procedure can be used in linear and nonlinear dynamic analysis.

In the proposed integration, time step is assumed to be constant.

The numerical integration is the heart of a dynamic analysis. The result's accuracy is strongly influenced by the accuracy and stability of the numerical integration.

This paper presents simple and accurate predictor‐corrector integration based on accelerations of several previous time steps. This may be used as a routine in any dynamic analysis software to enhance accuracy and reduce computational time.

Modelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.

Fourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.

Error magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 10³-10⁸, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.

The developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.

The proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

The purpose of this paper is to investigate the computational features of the Flowfield Dependent Variation (FDV) method, a numerical scheme built to simulate flows characterized by multiple speeds, multiple physical phenomena, and by large variations in flow variables.

Fundamentally, the FDV method may be regarded as a variant of the Lax-Wendroff Scheme (LWS) that is obtained by replacing the explicit time derivatives in LWS by a weighted combination of explicit and implicit time derivatives. The weighting factors – referred to as FDV parameters – may be broadly classified as convective and diffusive parameters which, for example, are determined using flow quantities such as the Mach number and Reynolds number, respectively. Hence, the reference to these parameters and the method as “flow field dependent.” A von Neumann Fourier analysis demonstrates that the increased implicitness makes FDV both more stable and less dispersive compared to LWS, a feature crucial to capturing shocks and other phenomena characterized by high gradients in variables. In the current study, the FDV scheme is implemented in a Taylor-Galerkin-based finite element method framework that supports arbitrarily high order, unstructured isoparametric elements in one-, two- and three-dimensional geometries.

At first, the spatial accuracy of the implemented FDV scheme is established using the Method of Manufactured Solutions, wherein the results show that the order of accuracy of the scheme is nearly equal to the order of the shape function polynomial plus one. The dispersion and dissipation errors of FDV, when applied to the compressible Navier-Stokes and energy equations, are investigated using a 2-D, small-amplitude acoustic pulse propagating in a quiescent medium. It is shown that FDV with third-order shape functions accurately captures both the amplitude and phase of the acoustic pulse. The method is then applied to cases ranging from low-Mach number subsonic flows (Mach number M=0.05) to high-Mach number supersonic flows (M=4) with shock-boundary layer interactions. For all cases, fair to good agreement is observed between the current results and those in the literature.

The spatial order of accuracy of the FDV method, its stability and dispersive properties, as well as its applicability to low- and high-Mach number flows are established.

In 1982, Smith and Hutton published comparative results of several different convection‐diffusion schemes applied to a specially devised test problem involving near‐discontinuities and strong streamline curvature. First‐order methods showed significant artificial diffusion, whereas higher‐order methods gave less smearing but had a tendency to overshoot and oscillate. Perhaps because unphysical oscillations are more obvious than unphysical smearing, the intervening period has seen a rise in popularity of low‐order artificially diffusive schemes, especially in the numerical heat‐transfer industry. This paper presents an alternative strategy of using non‐artificially diffusive higher‐order methods, while maintaining strictly monotonic transitions through the use of simple flux‐limiter constraints. Limited third‐order upwinding is usually found to be the most cost‐effective basic convection scheme. Tighter resolution of discontinuities can be obtained at little additional cost by using automatic adaptive stencil expansion to higher order in local regions, as needed.

This study aims to provide a reliable and effective algorithm that is suitable for addressing the problems of continuous orders of frequencies and modes under different boundary conditions, circumferential wave numbers and thickness-to-length ratios of moderately thick circular cylindrical shells. The theory of free vibration of rotating cylindrical shells is of utmost importance in fields such as structural engineering, rock engineering and aerospace engineering. The finite element method is commonly used to study the theory of free vibration of rotating cylindrical shells. The proposed adaptive finite element method can achieve a considerably more reliable high-precision solution than the conventional finite element method.

On a given finite element mesh, the solutions of the frequency mode of the moderately thick circular cylindrical shell were obtained using the conventional finite element method. Subsequently, the superconvergent patch recovery displacement method and high-order shape function interpolation techniques were introduced to obtain the superconvergent solution of the mode (displacement), while the superconvergent solution of the frequency was obtained using the Rayleigh quotient computation. Finally, the superconvergent solution of the mode was used to estimate the errors of the finite element solutions in the energy norm, and the mesh was subdivided to generate a new mesh in accordance with the errors.

In this study, a high-precision and reliable superconvergent patch recovery solution for the vibration modes of variable geometrical rotating cylindrical shells was developed. Compared with conventional finite element method, under the challenging varying geometrical circumferential wave numbers, and thickness–length ratios, the optimised finite element meshes and high-precision solutions satisfying the preset error limits were obtained successfully to solve the frequency and mode of continuous orders of rotating cylindrical shells with multiple boundary conditions such as simple and fixed supports, demonstrating good solution efficiency. The existing problem on the difficulty of adapting a set of meshes to the changes in vibration modes of different orders is finally overcome by applying the adaptive optimisation.

The approach developed in this study can accurately obtain the superconvergent patch recovery solution of the vibration mode of rotating cylindrical shells. It can potentially be extended to fine numerical models and high-precision computations of vibration modes (displacement field) and solid stress (displacement derivative field) for general structural special value problems, which can be extensively applied in the field of engineering computations in the future. Furthermore, the proposed method has the potential for adaptive analyses of shell structures and three-dimensional structures with crack damage. Compared with conventional finite element methods, significant advantages can be achieved by solving the eigenvalues of structures with high precision and stability.

This work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional time‐harmonic electromagnetic wave propagation problems.

The proposed HDG method for the discretization of the two‐dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element‐wise (local) solutions can be defined. A so‐called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux‐based DG method is also examined.

The approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L₂ norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux‐based DG method.

The work described here is a demonstration of the viability of a HDG formulation for solving the time‐harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.

The purpose of this paper is to present a pressure‐correction procedure for incompressible flows using unstructured meshes. A method of implementing high‐order spatial schemes on unstructured grids was introduced.

The procedure used a collocated cell‐centered unstructured grid arrangement. In order to improve the accuracy of calculation, the widely used high‐order schemes for convection, developed for structured grids and in the form of either the normalized variable and space formulation (NVSF) or the total variation diminishing (TVD) flux limiters (FL), were introduced and implemented onto the unstructured grids. This implementation was carried out by constructing a local coordinate and introducing a virtual upstream node.

The procedure was validated by calculating the lid‐driven cavity flows which had benchmark numerical solutions. For comparison, these flows were also computed by a commercial package, the FLUENT. The results obtained by the present procedure agreed well with the benchmark solution although very coarse grids were used. For the FLUENT, however, worse agreements with the benchmark solutions were obtained although the grids used for computation were the same. These demonstrated the robustness of the presented numerical procedure.

With the present method, high‐order schemes in either NVSF or TVD FL forms for structured grids can be easily implemented onto unstructured grids. This provides more choices of high‐order schemes for calculating complex flows.

The purpose of this paper is to investigate the two-phase flow problems involving gas–liquid mixture.

The governed equations consist of a range of conservation laws modeling a classification of two-phase flow phenomena subjected to a velocity nonequilibrium for the gas–liquid mixture. Effects of the relative velocity are accounted for in the present model by a kinetic constitutive relation coupled to a collection of specific equations governing mass and volume fractions for the gas phase. Unlike many two-phase models, the considered system is fully hyperbolic and fully conservative. The suggested relaxation approach switches a nonlinear hyperbolic system into a semilinear model that includes a source relaxation term and characteristic linear properties. Notably, this model can be solved numerically without the use of Riemann solvers or linear iterations. For accurate time integration, a high-resolution spatial reconstruction and a Runge–Kutta scheme with decreasing total variation are used to discretize the relaxation system.

The method is used in addressing various nonequilibrium two-phase flow problems, accompanied by a comparative study of different reconstructions. The numerical results demonstrate the suggested relaxation method’s high-resolution capabilities, affirming its proficiency in delivering accurate simulations for flow regimes characterized by strong shocks.

While relaxation methods exhibit notable performance and competitive features, as far as we are aware, there has been no endeavor to address nonequilibrium two-phase flow problems using these methods.