Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.

A systematic literature review of Burgers’ equation with recent advances

Steady- and unsteady-state theories for interfacial mass transfer are used to model gas absorption with second-order, reversible reaction and the absorption of CO2 into aqueous MDEA (methyldiethanolamine). The latter case represents a system of industrial interest, having both finite rate and equilibrium reactions. Steady-state theories studied are film theory, simplified eddy diffusivity theory, and an approximation to surface renewal theory. Higbie's penetration and Danckwert's surface renewal theories are the unsteady-state theories reviewed. The Nernst-Planck equations for diffusion and reaction in ionic systems are solved numerically using orthogonal collocation on finite elements. Comparisons are made between the absorption enhancement factor obtained for all theories. The applicability of approximate methods of solution for absorption with chemical reaction is also discussed.

Numerical simulation of theories for gas absorption with chemical reaction

ELSIM—a problem-solving environment for electrochemical kinetic simulations. Version 3.0-solution of governing equations associated with interfacial species, independent of spatial coordinates or in one-dimensional space geometry

https://pure.rug.nl/ws/files/3427889/1981JCompPhysBotta.pdf

On Local Relaxation Methods and Their Application to Convection-Diffusion Equations

The eigenvalue method, which has been used by researchers in structure mechanics, is applied to problems in heat conduction. Its formulation is decribed in terms of an examination of transient heat conduction in a square slab. Taking advantage of the availability of the exact solution, we compare the accuracy and other numerical properties of the eigenvalue method with those of existing numerical schemes. The comparsion shows that, overall, the eigenvalue method appears to be fairly attractive. Furthermore, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed and retained to yield reasonably high accuracy. Greater savings are attained in the computation time for a transient problem with long time duration and a large computational domain.

An eigenvalue method for solving transient heat conduction problems

A comparison of the Method of Lines to finite difference techniques in solving time-dependent partial differential equations

Abstract Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (i) Burger's equation over a finite space domain by a forward time—central space explicit method, and (ii) the stream function—vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to “set up” time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.

A comparison of the Method of Lines to finite difference techniques in solving time-dependent partial differential equations. [with applications to Burger equation and stream function-vorticity problem]

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A comparison of the Method of Lines to finite difference techniques in solving time-dependent partial differential equations

A comparison of the Method of Lines to finite difference techniques in solving time-dependent partial differential equations. [with applications to Burger equation and stream function-vorticity problem]