High-order compact finite difference schemes are used for solving third-order

differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

created during the study of obstacle boundary value problems. They have been shown to be highly accurate and efficient. They are constructed by modifying the second-order scheme that was developed by Noor and Al-Said in 2002. The convergence rate of the high-order compact scheme is third order, the second-order scheme is fourth order.

Differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

s are essential tools in

mathematical modelling A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

. Most

physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...

s are described in terms of mathematical models that include convective and diffusive transport of some variables.

Finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are dis ...

s are amongst the most popular methods that have been applied most frequently in solving such differential equations. A finite difference scheme is compact in the sense that the discretised formula comprises at most nine point

stencils Stencilling produces an image or pattern on a surface, by applying pigment to a surface through an intermediate object, with designed holes in the intermediate object, to create a pattern or image on a surface, by allowing the pigment to reach ...

which includes a

node In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex). Node may refer to: In mathematics * Vertex (graph theory), a vertex in a mathematical graph * Vertex (geometry), a point where two or more curves, line ...

in the middle about which differences are taken. In addition, greater order of accuracy (more than two) justifies the terminology 'higher-order compact finite difference scheme' (HOC). This can be achieved in several ways. The higher-order compact scheme considered here Kalita JC, Dalal DC and Dass AK., A class of higher-order compact schemes for the unsteady two-dimensional convection-diffusion equations with variable convection coefficients., Int. J. Numer. Meth. Fluids, Vol. 101,(2002), pp. 1111–1131 is by using the original differential equation to substitute for the leading

truncation error In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. Examples Infinite series A summation series for e^x is given by an infinite series such as e^x=1+ x+ \frac + \frac ...

terms in the finite difference equation. Overall, the scheme is found to be robust, efficient and accurate for most

computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate t ...

(CFD) applications discussed here further. The simplest problem for the validation of the numerical algorithms is the Lid Driven cavity problem. Computed results in form of tables, graphs and figures for a fluid with Prandtl number = 0.71 with Rayleigh number (Ra) ranging from 10³ to 10⁷ are available in the literature. The efficacy of the scheme is proved when it very clearly captures the secondary and tertiary vortices at the sides of the cavity at high values of Ra. Another milestone was the development of these schemes for solving two dimensional steady/unsteady convection diffusion equations. A comprehensive study of flow past an impulsively started circular cylinder was made. The problem of flow past a circular cylinder has continued to generate tremendous interest amongst researchers working in CFD mainly because it displays almost all the fluid mechanical phenomena for incompressible, viscous flows in the simplest of geometrical settings. It was able to analyze and visualize the flow patterns more accurately for Reynold's number (Re) ranging from 10 to 9500 compared to the existing numerical results. This was followed by its extension to rotating counterpart of the cylinder surface for Re ranging from 200 to 1000. More complex phenomenon that involves a circular cylinder undergoing rotational oscillations while translating in a fluid is studied for Re as high as 500. Another benchmark in the history is its extension to multiphase flow phenomena. Natural processes such as gas bubble in oil, ice melting, wet steam are observed everywhere in nature. Such processes also play an important role with the practical applications in the area of biology, medicine,

environmental remediation Environmental remediation deals with the removal of pollution or contaminants from environmental media such as soil, groundwater, sediment, or surface water. Remedial action is generally subject to an array of regulatory requirements, and may ...

. The scheme has been successively implemented to solve one and two dimensional

elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

and

parabolic equation A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivati ...

s with discontinuous coefficients and singular source terms. These type of problems hold importance numerically because they usually lead to non-smooth or discontinuous solutions across the interfaces. Expansion of this idea from fixed to moving interfaces with both regular and irregular geometries is currently going on. H. V. R. Mittal, Ray, Rajendra K. Ray, Solving Immersed Interface Problems Using a New Interfacial Points-Based Finite Difference Approach, SIAM Journal on Scientific Computing, vol. 40, no. 3 (2018), pp. A1860-A1883

References

{{reflist Finite differences Numerical differential equations