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The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

To develop a numerical technique for solving the inverse source problem associated with the constant coefficients convection‐diffusion equation.

The proposed numerical technique is based on the boundary element method (BEM) combined with an iterative sequential quadratic programming (SQP) procedure. The governing convection‐diffusion equation is transformed into a Helmholtz equation and the ill‐conditioned system of equations that arises after the application of the BEM is solved using an iterative technique.

The iterative BEM presented in this paper is well‐suited for solving inverse source problems for convection‐diffusion equations with constant coefficients. Accurate and stable numerical solutions were obtained for cases when the number of sources is correctly estimated, overestimated, or underestimated, and with both exact and noisy input data.

The proposed numerical method is limited to cases when the Péclet number is smaller than 100. Future approaches should include the application of the BEM directly to the convection‐diffusion equation.

Applications of the results presented in this paper can be of value in practical applications in both heat and fluid flow as they show that locations and strengths for an unknown number of point sources can be accurately found by using boundary measurements only.

The BEM has not as yet been employed for solving inverse source problems related with the convection‐diffusion equation. This study is intended to approach this problem by combining the BEM formulation with an iterative technique based on the SQP method. In this way, the many advantages of the BEM can be applied to inverse source convection‐diffusion problems.

The purpose of this paper is to reduce the computational burden and improve the precision of the parameter optimization in the convection-diffusion equation, a new algorithm, the refined gray-encoded evolution algorithm (RGEA), is proposed.

In the new algorithm, the differential evolution algorithm (DEA) is introduced to refine the solutions and to improve the search efficiency in the evolution process; the rapid cycle operation is also introduced to accelerate the convergence rate. The authors apply this algorithm to parameter optimization in convection-diffusion equations.

Two cases for parameter optimization in convection-diffusion equations are studied by using the new algorithm. The results indicate that the sum of absolute errors by the RGEA decreases from 74.14 to 99.29 percent and from 99.32 to 99.98 percent, respectively, compared to those by the gray-encoded genetic algorithm (GGA) and the DEA. And the RGEA has a faster convergent speed than does the GGA or DEA.

A more complete convergence analysis of the method is under investigation. The authors will also explore the possibility of adapting the method to identify the initial condition and boundary condition in high-dimension convection-diffusion equations.

This paper will have an important impact on the applications of the parameter optimization in the field of environmental flow analysis.

This paper will have an important significance for a sustainable social development.

The authors establish a new RGEA algorithm for parameter optimization in solving convection-diffusion equations. The application results make a valuable contribution to the parameter optimization in the field of environmental flow analysis.

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 paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

The paper presents the numerical performance of the preconditioned generalized conjugate gradient (PGCG) methods in solving non‐linear convection — diffusion equations. Three non‐linear systems which describe a non‐isothermal chemical reactor, the chemically driven convection in a porous medium and the incompressible steady flow past a sphere are the test problems. The standard second order accurate centred finite difference scheme is used to discretize the models equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the PGCG algorithm as inner iteration. Three PGCG techniques, which emerge to be the best performing, are tested. Laplace‐type operators are employed for preconditioning. The results show that the convergence of the PGCG methods depends strongly on the convection—diffusion ratio. The most robust algorithm is GMRES. But even with GMRES non‐convergence occurs when the convection—diffusion ratio exceeds a limit value. This value seems to be influenced by the non‐linearity type.

The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.

The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.

The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.

Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

This paper presents a boundary element formulation for transient convection‐diffusion problems employing the fundamental solution of the corresponding steady‐state equation with constant coefficients and a dual reciprocity approximation. The formulation allows the mathematical problem to be described in terms of boundary values only. Numerical results show that the BEM does not present oscillations or damping of the wave front as appear in other numerical techniques.

The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent convection-diffusion equations.

This paper opted for the time-dependent convection-diffusion equations using the finite element method and the partition of unity method.

This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method (SGM) but saves a lot of computational time.

In this paper, a parallel partition of unity method is proposed for the time-dependent convection-diffusion equations. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

The purpose of this paper is to find an effective numerical method for solving singularly perturbed convection-diffusion problems.

The present method is based on the asymptotic expansion method and the variational iteration method (VIM). First a zeroth order asymptotic expansion for the solution of the given singularly perturbed convection-diffusion problem is constructed. Then the reduced terminal value problem is solved by using the VIM.

Two numerical examples are introduced to show the validity of the present method. Obtained numerical results show that the present method can provide very accurate analytical approximate solutions not only in the boundary layer, but also away from the layer.

The combination of the asymptotic expansion method and the VIM is applied to singularly perturbed convection-diffusion problems.