A fourth-order method of the convection–diffusion equations with Neumann boundary conditions

A compact finite volume method and its extrapolation for elliptic equations with third boundary conditions

“…The treatment of the Neumann boundary conditions can be summarized into three categories. The first one uses divided central difference to discretize the derivatives at boundaries by introducing fictitious points on both sides of the domain [8][9][10] and the second one uses forward or backward difference to discretize the derivatives at boundaries [2,13]. Certainly, for these two methods, the necessary prerequisite to get high accuracy is that assuming the equation holds on the boundaries.…”

“…In the last decades, there have been great advances in developing high-order compact finite difference schemes for solving partial differential equations and a great deal of research work has been reported [2,4,[6][7][8][9][10][13][14][15]18,19]. Most existing high-order compact schemes are constructed for problems with Dirichlet boundary conditions [6,14] because Dirichlet boundary conditions are simple and straightforward to implement in a high-order compact stencil.…”

“…Most existing high-order compact schemes are constructed for problems with Dirichlet boundary conditions [6,14] because Dirichlet boundary conditions are simple and straightforward to implement in a high-order compact stencil. Recently, the derivation of high-order compact schemes for problems with Neumann boundary conditions has been becoming a topical research area [2,4,[8][9][10]13,15,18,19]. The treatment of the Neumann boundary conditions can be summarized into three categories.…”

Condition for Non-Oscillatory Solution for Scalar Convection-Dominated Equation

“…The choice of suitable computational grid to discretize the governing partial differential equations (e.g. by means of polynomial fitting, Taylor series expansion and compact scheme to obtain approximations to the derivatives of the variables with respect to the coordinates) is necessary at the onset of numerical modelling of the scalar convection-dominated problems as in some studies [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. It is worth to note here that the variable values at locations other than the defined grid nodes can also be determined by interpolation.…”

Novel Cubic Trigonometric B-Spline Approach Based on the Hermite Formula for Solving the Convection-Diffusion Equation

“…Restrictive Taylor approximation was used by Ismail et al [5] to solve the CDE. Cao et al [6] developed a fourth-order compact finite difference scheme for solving the CDE. e generalized trapezoidal formula was used by Chawla and Al-Zanaidi [7] to solve the CDE.…”