Higher order approximation in exponential form based on half-step grid-points for 2D quasilinear elliptic BVPs on a variant domain

Abstract

This paper reports a new fourth order Finite Difference Method (FDM) in exponential form for two-dimensional quasilinear boundary value problem of elliptic type (BVPE) with variant solution domain. Further, this discretization is extended to solve the system of quasilinear BVPEs. Following are the main highlights of the proposed FDM:

• An unequal mesh 9-point compact stencil is used to approximate the solution. Half-step points are used to evaluate the known variables of this problem. The convergence theory is studied for unequal mesh to validate the fourth order convergence of the suggested FDM.

• It is applicable to BVPE irrespective of coordinate systems. Various benchmark problems, for example, Poisson equation in cylindrical coordinates, Burgers’ equation, Navier-Stokes (NS) equations in cylindrical and rectangular coordinates, are solved to depict their fourth order convergence.

• Numerical results confirm the accuracy, trustworthiness and acceptability of the suggested numerical algorithm. These results endorse the superiority of the proposed FDM over the previously existing techniques of Mohanty and Kumar (2014), Mohanty and Setia (2014), Priyadarshini and Mohanty (2021).

Graphical abstract

Specifications table

Subject area;	Mathematics
More specific subject area;	Partial Differential Equations
Name of your method;	Finite Difference Technique
Name and reference of original method;	I. Priyadarshini, R.K. Mohanty High resolution half-step compact numerical approximation for 2D quasilinear elliptic equations in vector form and the estimates of normal derivatives on an irrational domain Soft Computing, 25 (2021) 9967–9991
Resource availability;	MATLAB

The Finite Difference Technique

Preliminaries and History

 

Introduction

We consider the following quasi-linear partial differential equation (PDE) of elliptic type:

$$\Phi \left(x,y,u\right)u_{xx}+\Psi \left(x,y,u\right)u_{yy}=R\left(x,y,u,u_x,u_y\right).$$ (1)

The solution domain for Eq. (1) is given by $\Omega =\left(0,x_{\alpha }\right)\times \left(0,y_{\beta }\right)$. Let its boundary be denoted by $\partial \Omega$.

The boundary values of Dirichlet type are prescribed as follows:

$$u\left(x,y\right)=u_0\left(x,y\right),\left(x,y\right)\in \partial \Omega ,$$ (2)

where real numbers $x_{\alpha }$ and $y_{\beta }$ are such that $\left(x_{\alpha }/y_{\beta }\right)$ is irrational. Eq. (1) along with the above prescribed boundary values (2) is termed as boundary value problem of elliptic type (BVPE).

It is supposed that Eq. (1) satisfies $\Phi .\Psi >0$ in $\Omega$, which is the condition for ellipticity.

For $\left(x,y\right)\in \Omega$, the following conditions hold:

• I. $u\in C^6\left(\Omega \right)$ and $\Phi ,\Psi \in C^4\left(\Omega \right)$,

• II. $R$ is $\text{continuous}$ and $\left(\frac{\partial \mathrm{R}}{\partial u}\right)\ge 0,$

• III. $|\frac{\partial \mathrm{R}}{\partial u_x}|\le K_1,\left|\frac{\partial \mathrm{R}}{\partial u_y}\right|\le K_2$, where $K_1$ and $K_2$ are constants in $\mathbb{R}$.

Condition (I) is vital for generation of local truncation error (LTE) of the required accuracy. (II)-(III) are the sufficient conditions required for the numerical technique to be convergent [1]. Further, it is assumed that the solution of the given BVPE (1)-(2) exists in the prescribed domain and is unique therein.

Associated research work done in the past

Over the span of last four decades, many numerical methods of high accuracy have been developed towards solving linear and non-linear BVPEs. A finite difference method (FDM) that involves only the grid points immediately surrounding the one in concern is called as a compact FDM. In this article, a compact FDM of order three has been presented and applied to solve the quasilinear BVPEs in vector form, such as the Navier-Stokes (NS) equations.

Some related research works done in the past by various authors are as follows: FDMs of different orders of approximation for solving linear and non-linear BVPEs have been developed by Jain et al. [7,8,10] and Mohanty [9,15,24]. In the year 1995, an FDM of order four for a two-dimensional (2D) mildly non-linear BVPE has been developed by Saldanha and Ananthakrishnaiah et al. [11]. That same year, the NS equations which form a system of non-linear BVPEs have been solved using SOR technique by Li et al. [12]. Further, Spotz and Carey [13,14], Tian and Ge [22] have formed a nine-point high order compact (HOC) FDM for stream-function vorticity form of time-independent NS equations. In 1997, fourth order approximations have been given for a significant linear counterpart of the given BVPE (1) – (2) by Yavneh [16] and Zhang [17]. Further, compact techniques have been designed for 2D BVPEs and the convergence of the iterative techniques involved have been discussed by Zhang [18] and Saldanha [21]. In the year 1999, an HOC FDM has been proposed by Yanwen et al. [19] for solving incompressible system of NS equations in a regular domain. Mohanty and Dey [20] have framed approximations of order four, towards finding the normal derivatives of the dependent variable involved in a quasi-linear BVPE. In 2006, another new compact discretization for NS equations has been given by Erturk and Gökcöl [23]. They have used a finer mesh and a tri-diagonal solver. The order of accuracy achieved is four. Preconditioning techniques have been followed by Arabshahi and Dehghan [25] to solve a linear counterpart of the BVPE in concern. In 2008, Liu and Wang [26] have developed an FDM towards solving the mean-vorticity formulation, and examined the computational results obtained by application to equations governing large-scale atmospheric and oceanic flow. That same year, Fairweather et al. [27] proposed a discretization using orthogonal spline collocation method for 2D NS equations using stream-function vorticity form. Further, Ito and Qiao [28] have used a new HOC FDM to solve a BVPE involving the Stokes equations subject to Dirichlet boundary conditions. Upwind compact estimates of high orders for NS equations involving artificial compressibility have been designed by Shah and Yuan [31]. Apart from these, some finite element methods (FEM) ([6,29]) and finite volume methods [30] have also been proposed for the solution of many analogous forms of the BVPE in concern.

Detailing the related studies done in past ten years: In 2011, Tian et al. [32] have proposed a compact discretization of high order for incompressible NS equations. During the subsequent years, Mohanty and Setia have developed an HOC FDM for a general system of quasi-linear BVPEs using off-step grid points with uniform mesh in [33] and an unequal mesh in [36]. A scalar counterpart of this problem of Numerov type has been solved by Mohanty and Kumar [35]. Collocation methods based on Haar and Legendre wavelets for solving PDEs of elliptic type have been given by Aziz et al. [60] in 2013. Further, in the year 2014, a new HOC FDM for a Poisson equation in 2D space has been designed by Zhai et al. [34]. A comparison of an FDM and an FEM applied over a linear BVPE has been studied and discussed by Papanikos and Gousidou-Koutita in [37]. Monotone and adaptive FDMs have been combined to form a new FDM by Oberman and Zwiers [38] for time-dependent and time-independent problems with free boundaries. A similar method is designed by Hamfeldt and Salvador [43]. In 2015, Islam et al. [61] have developed a collocation method and a meshless method for 2D Poisson equation with nonlocal boundary conditions. Various meshless techniques have been derived by many authors in the subsequent years ([40,41,50,51]). Another FDM applied to NS equations is designed by Bayona et al. [42]. In 2017, Aziz et al. [62] have developed a collocation technique for a general three dimensional (3D) BVPE. Further, a class of FDMs with effective application to time-independent as well as time-dependent NS equations has been developed by Mittal et al. [39]. Subsequently, in [44], Mittal and Ray have developed FDMs for non-linear BVPEs in two- and three-dimensional space. Poisson equation with variable coefficients has been solved by Raeli et al. in [45]. Pandey and Jaboob [46] have proposed an FDM of order two, for a system involving BVPEs. Further, Zhang et al. [47] have solved 2D NS equations using a Galerkin method. Nadeem and Aziz [63] have designed a meshless and a Haar wavelet collocation method for 2D and 3D PDEs of elliptic type. In the year 2020, Li and Zhang [52] have developed a new HOC FDM by reducing the classical FEM. HOC FDMs in exponential form have been recently developed by Mohanty et al. [48,49,53,54] and Manchanda et al. [55]. Most recently, new high order approximations for 2D quasi-linear BVPEs involving an unequal mesh have been given by Priyadarshini and Mohanty in [56] and [57]. Mohanty et al. [59] have presented an HOC FDM in exponential form for the concerned system of BVPEs using full step grid points.

In the present article, we propose a new HOC FDM in exponential form using an unequal mesh for the given BVPE (1) – (2) on a dissimilar domain. The discretization so developed, involves nine points of the compact cell. The method details comprise of the mesh formation, notations involved for framing the proposed method, the discretization and its derivation. Further, we give a proof of the order of the discretization through a convergence analysis. The order of accuracy of the proposed FDM has been validated with computational illustrations through many benchmark BVPEs.

Numerical scheme on unequal mesh

For the construction of the new HOC FDM, we consider the 2D PDE:

$$\Phi \left(x,y\right)u_{xx}+\Psi \left(x,y\right)u_{yy}=R\left(x,y,u,u_x,u_y\right),\left(x,y\right)\in \Omega .$$ (3)

Let us impose a mesh on the domain using the grid sizes $h_x>0$ and $h_y>0$ in x- and y- directions, respectively, such that $h_x\ne h_y$ in general. The mesh points in respective directions are given by $x_i=i{h}_x,i=0,1,\dots ,N_x$, and $y_j=j{h}_y,j=0,1,\dots ,N_y$, where $N_x$ and $N_y$ are positive integers such that $N_x{h}_x=x_{\alpha }$,$N_y{h}_y=y_{\beta }.$ Let$U_{i,j}=u\left(x_i,y_j\right)$ and is approximated by $u_{i,j}$. Let $\gamma =\left(\frac{h_x}{h_y}\right)>0$ and ${\Phi }_{i,j}=\Phi \left(x_i,y_j\right)$, ${\Psi }_{i,j}=\Psi \left(x_i,y_j\right)$.

At each nodal point $\left(x_i,y_j\right)$, let the following notations hold:

$${\mathrm{U}}_{\zeta \eta }=\frac{{\partial }^{\zeta +\eta }\mathrm{U}}{\partial x^{\zeta }\partial y^{\eta }},{\Phi }_{\zeta \eta }=\frac{{\partial }^{\zeta +\eta }\Phi }{\partial x^{\zeta }\partial y^{\eta }},{\Psi }_{\zeta \eta }=\frac{{\partial }^{\zeta +\eta }\Psi }{\partial x^{\zeta }\partial y^{\eta }}\text{for}\zeta ,\eta =0,1,2,\dots$$ (4)

$$q_1=\frac{-h_x{\Phi }_{10}}{6{\Phi }_{00}},q_2=\frac{-h_y{\Psi }_{01}}{6{\Psi }_{00}}.$$ (5)

Further,

$$U_0=U_{i,j},U_1=U_{i+1,j},U_2=U_{i-1,j},U_3=U_{i,j+1},U_4=U_{i,j-1},U_5=U_{i+1,j+1},U_6=U_{i+1,j-1},U_7=U_{i-1,j+1},U_8=U_{i-1,j-1}.$$ (6)

For the domain $\Omega$, let

$${\widehat{U}}_{i+\frac{1}{2},j}=\frac{1}{2}\left(U_0+U_1\right),$$ (7)

$${\widehat{U}}_{i-\frac{1}{2},j}=\frac{1}{2}\left(U_0+U_2\right),$$ (8)

$${\widehat{U}}_{i,j+\frac{1}{2}}=\frac{1}{2}\left(U_0+U_3\right),$$ (9)

$${\widehat{U}}_{i,j-\frac{1}{2}}=\frac{1}{2}\left(U_0+U_4\right),$$ (10)

$${\widehat{U}}_{xi,j}=\frac{1}{2h_x}\left(U_1-U_2\right),$$ (11)

$${\widehat{U}}_{xi+\frac{1}{2},j}=\frac{1}{h_x}\left(U_1-U_0\right),$$ (12)

$${\widehat{U}}_{xi-\frac{1}{2},j}=\frac{1}{h_x}\left(U_0-U_2\right),$$ (13)

$${\widehat{U}}_{xi,j+\frac{1}{2}}=\frac{1}{4h_x}\left(U_5-U_7+U_1-U_2\right),$$ (14)

$${\widehat{U}}_{xi,j-\frac{1}{2}}=\frac{1}{4h_x}\left(U_6-U_8+U_1-U_2\right),$$ (15)

$${\widehat{U}}_{yi,j}=\frac{1}{2h_y}\left(U_3-U_4\right),$$ (16)

$${\widehat{U}}_{yi+\frac{1}{2},j}=\frac{1}{4h_y}\left(U_5-U_6+U_3-U_4\right),$$ (17)

$${\widehat{U}}_{yi-\frac{1}{2},j}=\frac{1}{4h_y}\left(U_7-U_8+U_3-U_4\right),$$ (18)

$${\widehat{U}}_{yi,j+\frac{1}{2}}=\frac{1}{h_y}\left(U_3-U_0\right),$$ (19)

$${\widehat{U}}_{yi,j-\frac{1}{2}}=\frac{1}{h_y}\left(U_0-U_4\right),$$ (20)

$${\widehat{U}}_{xxi,j}=\frac{1}{h_x^2}\left(U_1+U_2-2U_0\right),$$ (21)

$${\widehat{U}}_{yyi,j}=\frac{1}{h_y^2}\left(U_3+U_4-2U_0\right),$$ (22)

$${\widehat{U}}_{xxyi,j}=\frac{1}{2h_x^2{h}_y}\left[U_5-U_6+U_7-U_8-2\left(U_3-U_4\right)\right],$$ (23)

$${\widehat{U}}_{xyyi,j}=\frac{1}{2h_y^2{h}_x}\left[U_5+U_6-U_7-U_8-2\left(U_1-U_2\right)\right],$$ (24)

$${\widehat{U}}_{xxyyi,j}=\frac{1}{h_x^2{h}_y^2}\left[U_5+U_6+U_7+U_8-2\left(U_1+U_2+U_3+U_4\right)+4U_0\right],$$ (25)

$${\widehat{\mathrm{R}}}_{i\pm \frac{1}{2},j}=R\left(x_{i\pm \frac{1}{2}},y_j,{\widehat{U}}_{i\pm \frac{1}{2},j},{\widehat{U}}_{xi\pm \frac{1}{2},j},{\widehat{U}}_{yi\pm \frac{1}{2},j}\right),$$ (26)

$${\widehat{R}}_{i,j\pm \frac{1}{2}}=R\left(x_i,y_{j\pm \frac{1}{2}},{\widehat{U}}_{i,j\pm \frac{1}{2}},{\widehat{U}}_{xi,j\pm \frac{1}{2}},{\widehat{U}}_{yi,j\pm \frac{1}{2}}\right),$$ (27)

$${\widehat{\widehat{U}}}_{xi,j}={\widehat{U}}_{xi,j}-\frac{h_x}{6{\Phi }_{00}}\left({\widehat{R}}_{i+\frac{1}{2},j}-{\widehat{R}}_{i-\frac{1}{2},j}\right)+\frac{h_x^2}{6{\Phi }_{00}}\left[{\Psi }_{00}{\widehat{U}}_{xyyi,j}+{\Phi }_{10}{\widehat{U}}_{xxi,j}+{\Psi }_{10}{\widehat{U}}_{yyi,j}\right],$$ (28)

$${\widehat{\widehat{U}}}_{yi,j}={\widehat{U}}_{yi,j}-\frac{h_y}{6{\Psi }_{00}}\left({\widehat{R}}_{i,j+\frac{1}{2}}-{\widehat{R}}_{i,j-\frac{1}{2}}\right)+\frac{h_y^2}{6{\Psi }_{00}}\left[{\Phi }_{00}{\widehat{U}}_{xxyi,j}+{\Phi }_{01}{\widehat{U}}_{xxi,j}+{\Psi }_{01}{\widehat{U}}_{yyi,j}\right],$$ (29)

$${\tilde{\tilde{U}}}_{i,j}=U_{i,j}+\frac{h_x^2}{16}{\widehat{U}}_{xxi,j}+\frac{h_y^2}{16}{\widehat{U}}_{yyi,j},$$ (30)

$${\tilde{\tilde{U}}}_{xi,j}={\widehat{U}}_{xi,j}-\frac{h_x}{16{\Phi }_{00}}\left({\widehat{R}}_{i+\frac{1}{2},j}-{\widehat{R}}_{i-\frac{1}{2},j}\right)+\frac{h_x^2}{16}\left[\frac{1}{{\gamma }^2}+\frac{{\Psi }_{00}}{{\Phi }_{00}}\right]{\widehat{U}}_{xyyi,j}+\frac{h_x^2}{16{\Phi }_{00}}\left[{\Phi }_{10}{\widehat{U}}_{xxi,j}+{\Psi }_{10}{\widehat{U}}_{yyi,j}\right],$$ (31)

$${\tilde{\tilde{U}}}_{yi,j}={\widehat{U}}_{yi,j}-\frac{h_y}{16{\Psi }_{00}}\left({\widehat{R}}_{i,j+\frac{1}{2}}-{\widehat{R}}_{i,j-\frac{1}{2}}\right)+\frac{h_y^2}{16}\left[{\gamma }^2+\frac{{\Phi }_{00}}{{\Psi }_{00}}\right]{\widehat{U}}_{xxyi,j}+\frac{h_y^2}{16{\Psi }_{00}}\left[{\Phi }_{01}{\widehat{U}}_{xxi,j}+{\Psi }_{01}{\widehat{U}}_{yyi,j}\right],$$ (32)

$${\widehat{\widehat{R}}}_{i,j}=R\left(x_i,y_j,U_{i,j},{\widehat{\widehat{U}}}_{xi,j},{\widehat{\widehat{U}}}_{yi,j}\right),$$ (33)

$${\tilde{\tilde{R}}}_{i,j}=R\left(x_i,y_j,{\tilde{\tilde{U}}}_{i,j},{\tilde{\tilde{U}}}_{xi,j},{\tilde{\tilde{U}}}_{yi,j}\right).$$ (34)

With the above definitions, at every $\left(x_i,y_j\right)$ inside $\Omega$, the PDE (3) is discretized by

$$\begin{array}{ccc}\hfill L_u& \equiv & [{\Phi }_{00}+q_1{h}_x{\Phi }_{10}+q_2{h}_y{\Phi }_{01}+\frac{h_x^2}{12}{\Phi }_{20}+\frac{h_y^2}{12}{\Phi }_{02}]{\widehat{U}}_{\mathit{xx}i,j}+[{\Psi }_{00}+q_1{h}_x{\Psi }_{10}+q_2{h}_y{\Psi }_{01}+\frac{h_x^2}{12}{\Psi }_{20}+\frac{h_y^2}{12}{\Psi }_{02}]{\widehat{U}}_{\mathit{yy}i,j}\hfill \\ & & +[q_2{h}_y{\Phi }_{00}+\frac{h_y^2}{6}{\Phi }_{01}]{\widehat{U}}_{\mathit{xxy}i,j}+[q_1{h}_x{\Psi }_{00}+\frac{h_x^2}{6}{\Psi }_{10}]{\widehat{U}}_{\mathit{xyy}i,j}+\frac{1}{12}[h_x^2{\Psi }_{00}+h_y^2{\Phi }_{00}]{\widehat{U}}_{\mathit{xxyy}i,j}\hfill \\ & =& {\widehat{\widehat{\mathrm{R}}}}_{i,j}\exp \left[\frac{(1+3q_1){\widehat{\mathrm{R}}}_{i+\frac{1}{2},j}+(1-3q_1){\widehat{\mathrm{R}}}_{i-\frac{1}{2},j}+(1+3q_2){\widehat{\mathrm{R}}}_{i,j+\frac{1}{2}}+(1-3q_2){\widehat{\mathrm{R}}}_{i,j-\frac{1}{2}}-4{\widetilde{\tilde{\mathrm{R}}}}_{i,j}}{3{\widehat{\widehat{\mathrm{R}}}}_{i,j}}\right]+{\widehat{\mathrm{T}}}_{i,j},\hfill \end{array}$$ (35)

where ${\widehat{\mathrm{T}}}_{i,j}=O\left(h_x^4+h_x^2{h}_y^2+h_y^4\right)$ is the LTE.

Using the prescribed Dirichlet values (2), the scheme (35) is reducible as a block matrix of tri-diagonal form. This system of equations is implicitly solvable by the point or block iterative technique [[2], [3], [3], [4], [5], [6]].

Derivation of the proposed FDM

The PDE (3) can be written as

$${\Phi }_{00}{U}_{20}+{\Psi }_{00}{U}_{02}=R\left(x_i,y_j,U_{i,j},U_{xi,j},U_{yi,j}\right)=R_{i,j},$$ (36)

at each internal grid point $\left(x_i,y_j\right).$

From (36), it is easy to observe that

$$\begin{array}{ccc}\hfill L_u& \equiv & [{\Phi }_{00}+q_1{h}_x{\Phi }_{10}+q_2{h}_y{\Phi }_{01}+\frac{h_x^2}{12}{\Phi }_{20}+\frac{h_y^2}{12}{\Phi }_{02}]{\widehat{U}}_{\mathit{xx}i,j}+[{\Psi }_{00}+q_1{h}_x{\Psi }_{10}+q_2{h}_y{\Psi }_{01}+\frac{h_x^2}{12}{\Psi }_{20}+\frac{h_y^2}{12}{\Psi }_{02}]{\widehat{U}}_{\mathit{yy}i,j}\hfill \\ & & +[q_2{h}_y{\Phi }_{00}+\frac{h_y^2}{6}{\Phi }_{01}]{\widehat{U}}_{\mathit{xxy}i,j}+[q_1{h}_x{\Psi }_{00}+\frac{h_x^2}{6}{\Psi }_{10}]{\widehat{U}}_{\mathit{xyy}i,j}+\frac{1}{12}[h_x^2{\Psi }_{00}+h_y^2{\Phi }_{00}]{\widehat{U}}_{\mathit{xxyy}i,j}\hfill \\ & =& R_{i,j}\exp \left[\frac{(1+3q_1)R_{i+\frac{1}{2},j}+(1-3q_1)R_{i-\frac{1}{2},j}+(1+3q_2)R_{i,j+\frac{1}{2}}+(1-3q_2)R_{i,j-\frac{1}{2}}-4R_{i,j}}{3R_{i,j}}\right]+O(h_x^4+h_x^2{h}_y^2+h_y^4).\hfill \end{array}$$ (37)

The new simplified version of approximations (7)–(25) can be written as

$${\widehat{U}}_{i\pm \frac{1}{2},j}=U_{i\pm \frac{1}{2},j}+\frac{h_x^2}{8}{U}_{20}\pm O\left(h_x^3\right),$$ (38)

$${\widehat{U}}_{i,j\pm \frac{1}{2}}=U_{i,j\pm \frac{1}{2}}+\frac{h_y^2}{8}{U}_{02}\pm O\left(h_y^3\right),$$ (39)

$${\widehat{U}}_{xi,j}=U_{xi,j}+\frac{h_x^2}{6}{U}_{30}+O\left(h_x^4\right),$$ (40)

$${\widehat{U}}_{xi\pm \frac{1}{2},j}=U_{xi\pm \frac{1}{2},j}+\frac{h_x^2}{24}{U}_{30}\pm O\left(h_x^3\right),$$ (41)

$${\widehat{U}}_{xi,j\pm \frac{1}{2}}=U_{xi,j\pm \frac{1}{2}}+\frac{h_x^2}{6}{U}_{30}+\frac{h_y^2}{8}{U}_{12}\pm O\left(h_x^2{h}_y\right),$$ (42)

$${\widehat{U}}_{yi,j}=U_{yi,j}+\frac{h_y^2}{6}{U}_{03}+O\left(h_y^4\right),$$ (43)

$${\widehat{U}}_{yi\pm \frac{1}{2},j}=U_{yi\pm \frac{1}{2},j}+\frac{h_x^2}{8}{U}_{21}+\frac{h_y^2}{6}{U}_{03}\pm O\left(h_x{h}_y^2\right),$$ (44)

$${\widehat{U}}_{yi,j\pm \frac{1}{2}}=U_{yi,j\pm \frac{1}{2}}+\frac{h_y^2}{24}{U}_{03}\pm O\left(h_y^3\right),$$ (45)

$${\widehat{U}}_{xxi,j}=U_{xxi,j}+\frac{h_x^2}{12}{U}_{40}+O\left(h_x^4\right),$$ (46)

$${\widehat{U}}_{yyi,j}=U_{yyi,j}+\frac{h_y^2}{12}{U}_{04}+O\left(h_y^4\right),$$ (47)

$${\widehat{U}}_{xxyi,j}=U_{xxyi,j}+O\left(h_x^2+h_y^2\right),$$ (48)

$${\widehat{U}}_{xyyi,j}=U_{xyyi,j}+O\left(h_x^2+h_y^2\right),$$ (49)

$${\widehat{U}}_{xxyyi,j}=U_{xxyyi,j}+O\left(h_x^2+h_y^2\right).$$ (50)

Executing the approximations (38)–(45) in (26), (27), we get

$${\widehat{\mathrm{R}}}_{i\pm \frac{1}{2},j}={\mathrm{R}}_{i\pm \frac{1}{2},j}+\frac{h_x^2}{24}{T}_1+\frac{h_y^2}{6}{U}_{03}{\mathrm{R}}_{u_y}\pm O\left(h_x^3+h_x{h}_y^2\right),$$ (51)

$${\widehat{\mathrm{R}}}_{i,j\pm \frac{1}{2}}={\mathrm{R}}_{i,j\pm \frac{1}{2}}+\frac{h_x^2}{6}{U}_{30}{R}_{u_x}+\frac{h_y^2}{24}{T}_2\pm O\left(h_y^3+h_y{h}_x^2\right),$$ (52)

where

$$T_1=3U_{20}{R}_u+U_{30}{R}_{u_x}+3U_{21}{R}_{u_y},$$

$$T_2=3U_{02}{R}_u+3U_{12}{R}_{u_x}+U_{03}{R}_{u_y}.$$

In order to obtain higher order approximations for $u_x$ and $u_y$, let

$${\widehat{\widehat{U}}}_{xi,j}={\widehat{U}}_{xi,j}+b_{11}{h}_x\left({\widehat{\mathrm{R}}}_{i+\frac{1}{2},j}-{\widehat{\mathrm{R}}}_{i-\frac{1}{2},j}\right)+h_x^2\left(b_{12}{\widehat{U}}_{xyyi,j}+b_{13}{\widehat{U}}_{xxi,j}+b_{14}{\widehat{U}}_{yyi,j}\right),$$ (53)

$${\widehat{\widehat{U}}}_{yi,j}={\widehat{U}}_{yi,j}+c_{11}{h}_y\left({\widehat{\mathrm{R}}}_{i,j+\frac{1}{2}}-{\widehat{\mathrm{R}}}_{i,j-\frac{1}{2}}\right)+h_y^2\left(c_{12}{\widehat{U}}_{xxyi,j}+c_{13}{\widehat{U}}_{xxi,j}+c_{14}{\widehat{U}}_{yyi,j}\right),$$ (54)

where $b_{1k},c_{1k}$, $k=$1,2,3,4 are parameters to be evaluated. Using the approximations (40), (43), (46)–(52), in (53), (54), we get

$${\widehat{\widehat{U}}}_{xi,j}=U_{xi,j}+\frac{h_x^2}{6}{T}_3+O\left(h_x^4+h_x^2{h}_y^2\right),$$ (55)

$${\widehat{\widehat{U}}}_{yi,j}=U_{yi,j}+\frac{h_y^2}{6}{T}_4+O\left(h_y^4+h_x^2{h}_y^2\right),$$ (56)

where

$$T_3=\left(1+6b_{11}{\Phi }_{00}\right)U_{30}+6\left(b_{11}{\Psi }_{00}+b_{12}\right)U_{12}+6\left(b_{11}{\Phi }_{10}+b_{13}\right)U_{20}+6\left(b_{11}{\Psi }_{10}+b_{14}\right)U_{02},$$

$$T_4=\left(1+6c_{11}{\Psi }_{00}\right)U_{03}+6\left(c_{11}{\Phi }_{00}+c_{12}\right)U_{21}+6\left(c_{11}{\Phi }_{01}+c_{13}\right)U_{20}+6\left(c_{11}{\Psi }_{01}+c_{14}\right)U_{02}.$$

Let $T_3=T_4=0$, then the values of parameters are determined by

$$b_{11}=\frac{-1}{6{\Phi }_{00}},b_{12}=\frac{{\Psi }_{00}}{6{\Phi }_{00}},b_{13}=\frac{{\Phi }_{10}}{6{\Phi }_{00}},b_{14}=\frac{{\Psi }_{10}}{6{\Phi }_{00}}$$

$$c_{11}=\frac{-1}{6{\Psi }_{00}},c_{12}=\frac{{\Phi }_{00}}{6{\Psi }_{00}},c_{13}=\frac{{\Phi }_{01}}{6{\Psi }_{00}},c_{14}=\frac{{\Psi }_{01}}{6{\Psi }_{00}}.$$

The approximations (55), (56) reduce to

$${\widehat{\widehat{U}}}_{xi,j}=U_{xi,j}+O\left(h_x^4+h_x^2{h}_y^2\right),$$ (57)

$${\widehat{\widehat{U}}}_{yi,j}=U_{yi,j}+O\left(h_y^4+h_x^2{h}_y^2\right).$$ (58)

With the aid of the approximations (57), (58), from (33), we have

$${\widehat{\widehat{R}}}_{i,j}=R_{i,j}+O\left(h_x^4+h_x^2{h}_y^2+h_y^4\right)$$ (59)

In order to increase the order of accuracy of the suggested method, we consider the following linear combinations:

$${\tilde{\tilde{U}}}_{i,j}=U_{i,j}+a_{21}{h}_x^2{\widehat{U}}_{xxi,j}+a_{22}{h}_y^2{\widehat{U}}_{yyi,j},$$ (60)

$${\tilde{\tilde{U}}}_{xi,j}={\widehat{U}}_{xi,j}+b_{21}{h}_x\left({\widehat{\mathrm{R}}}_{i+\frac{1}{2},j}-{\widehat{\mathrm{R}}}_{i-\frac{1}{2},j}\right)+h_x^2\left(b_{22}{\widehat{U}}_{xyyi,j}+b_{23}{\widehat{U}}_{xxi,j}+b_{24}{\widehat{U}}_{yyi,j}\right),$$ (61)

$${\tilde{\tilde{U}}}_{yi,j}={\widehat{U}}_{yi,j}+c_{21}{h}_y\left({\widehat{\mathrm{R}}}_{i,j+\frac{1}{2}}-{\widehat{\mathrm{R}}}_{i,j-\frac{1}{2}}\right)+h_y^2\left(c_{22}{\widehat{U}}_{xxyi,j}+c_{23}{\widehat{U}}_{xxi,j}+c_{24}{\widehat{U}}_{yyi,j}\right),$$ (62)

where $a_{21}$, $a_{22}$, $b_{2k},c_{2k}$, $k=$1,2,3,4 are variable parameters to be found out. With the assistance of (40), (43), (46)–(52), simplifying (60)–(62), we get

$${\tilde{\tilde{U}}}_{i,j}=U_{i,j}+a_{21}{h}_x^2{U}_{xxi,j}+a_{22}{h}_y^2{U}_{yyi,j}+O\left(h_x^4+h_y^4\right),$$ (63)

$${\tilde{\tilde{U}}}_{xi,j}=U_{xi,j}+\frac{h_x^2}{6}{T}_5+O\left(h_x^4+h_x^2{h}_y^2\right),$$ (64)

$${\tilde{\tilde{U}}}_{yi,j}=U_{yi,j}+\frac{h_y^2}{6}{T}_6+O\left(h_y^4+h_x^2{h}_y^2\right),$$ (65)

where

$$T_5=\left(1+6b_{21}{\Phi }_{00}\right)U_{30}+6\left(b_{21}{\Psi }_{00}+b_{22}\right)U_{12}+6\left(b_{21}{\Phi }_{10}+b_{23}\right)U_{20}+6\left(b_{21}{\Psi }_{10}+b_{24}\right)U_{02},$$

$$T_6=\left(1+6c_{21}{\Psi }_{00}\right)U_{03}+6\left(c_{21}{\Phi }_{00}+c_{22}\right)U_{21}+6\left(c_{21}{\Phi }_{01}+c_{23}\right)U_{20}+6\left(c_{21}{\Psi }_{01}+c_{24}\right)U_{02}.$$

As a result of (63)–(65), from (34), we have

$${\widetilde{\tilde{\mathrm{R}}}}_{i,j}={\mathrm{R}}_{i,j}+\left(a_{21}{h}_x^2{U}_{20}+a_{22}{h}_y^2{U}_{02}\right)R_u+\frac{h_x^2}{6}{T}_5{R}_{u_x}+\frac{h_y^2}{6}{T}_6{R}_{u_y}+O\left(h_x^4+h_x^2{h}_y^2+h_y^4\right).$$ (66)

By the support of Eqs. (51)–(52), (59) and (66), from (35) to (37), we may acquire the following LTE:

$$\begin{array}{ccc}\hfill {\widehat{\mathrm{T}}}_{i,j}& =& -h_x^2(1-16a_{21})U_{20}{R}_u-h_y^2(1-16a_{22})U_{02}{R}_u\hfill \\ & & +h_x^2[(1+16b_{21}{\Phi }_{00})U_{30}+16(b_{21}{\Phi }_{10}+b_{23})U_{20}+16(b_{21}{\Psi }_{10}+b_{24})U_{02}]R_{u_x}\hfill \\ & & -h_y^2[1-16{\gamma }^2(b_{21}{\Psi }_{00}+b_{22})]U_{12}{R}_{u_x}-h_y^2[{\gamma }^2-16(c_{21}{\Phi }_{00}+c_{22})]U_{21}{R}_{u_y}\hfill \\ & & +h_y^2[(1+16c_{21}{\Psi }_{00})U_{03}+16(c_{21}{\Phi }_{01}+c_{23})U_{20}+16(c_{21}{\Psi }_{01}+c_{24})U_{02}]R_{u_y}\hfill \\ & & +O(h_x^4+h_x^2{h}_y^2+h_y^4).\hfill \end{array}$$ (67)

We observe that to achieve the required accuracy, the coefficients of $h_x^2$ and $h_y^2$ in (67) must be equated to zero. This leads to ${\widehat{\mathrm{T}}}_{i,j}=O\left(h_x^4+h_x^2{h}_y^2+h_y^4\right)$ in the domain $\Omega$. Consequently, we obtain

$$a_{21}=a_{22}=\frac{1}{16},b_{21}=\frac{-1}{16{\Phi }_{00}},b_{22}=\frac{\left({\Phi }_{00}+{\gamma }^2{\Psi }_{00}\right)}{16{\gamma }^2{\Phi }_{00}},b_{23}=\frac{{\Phi }_{10}}{16{\Phi }_{00}},b_{24}=\frac{{\Psi }_{10}}{16{\Phi }_{00}},$$

$$c_{21}=\frac{-1}{16{\Psi }_{00}},c_{22}=\frac{\left({\Phi }_{00}+{\gamma }^2{\Psi }_{00}\right)}{16{\Psi }_{00}},c_{23}=\frac{{\Phi }_{01}}{16{\Psi }_{00}},c_{24}=\frac{{\Psi }_{01}}{16{\Psi }_{00}}.$$

The above technique given by (35) in combination with that of [57] will lead to the FDM of order four for the quasilinear counterpart of (1). This technique so developed can be extended to produce an $O\left(h_x^4+h_x^2{h}_y^2+h_y^4\right)$ approximation for BVPEs in vector-matrix system (see [33]).

The foremost advantage of the suggested technique is that it is appropriate to solve BVPEs irrespective of the coordinates system involved. No special technique or modification is required in order to obtain fourth order numerical solution near the singular points.

Convergence theory

Consider the following equation

$$\Phi u_{xx}+\Psi u_{yy}=R\left(x,y,u,u_x,u_y\right),$$ (68)

with prescribed Dirichlet boundary condition (2), assuming $\Phi ,\Psi$ to be constant values. Let $\gamma =\left(\frac{h_x}{h_y}\right)=$ a positive constant. We now show that the FDM (35) is $O\left(h_x^4\right)$ convergent when applied to (68). Throughout this subsection, $i,j$ are integers belonging to the specified intervals.

For $i,j=\left[1,N_x-1\right]\cap \mathbb{Z}$, at every $\left(x_i,y_j\right)$, the proposed FDM (35) applied over Eq. (68) results in the following:

$$\begin{array}{ccc}& & \left(\frac{\Phi +{\gamma }^2\Psi }{2}\right)U_8-\left(\Phi -5{\gamma }^2\Psi \right)U_4+\left(\frac{\Phi +{\gamma }^2\Psi }{2}\right)U_6-\left({\gamma }^2\Psi -5\Phi \right)U_2-10\left(\Phi +{\gamma }^2\Psi \right)U_0\hfill \\ & & -\left({\gamma }^2\Psi -5\Phi \right)U_1+\left(\frac{\Phi +{\gamma }^2\Psi }{2}\right)U_7-\left(\Phi -5{\gamma }^2\Psi \right)U_3+\left(\frac{\Phi +{\gamma }^2\Psi }{2}\right)U_5\hfill \\ & & =2h_x^2\left[3{\widehat{\widehat{\mathrm{R}}}}_{i,j}+{\widehat{\mathrm{R}}}_{i+\frac{1}{2},j}+{\widehat{\mathrm{R}}}_{i-\frac{1}{2},j}+{\widehat{\mathrm{R}}}_{i,j+\frac{1}{2}}+{\widehat{\mathrm{R}}}_{i,j-\frac{1}{2}}-4{\widetilde{\tilde{\mathrm{R}}}}_{i,j}\right]+{\widehat{T}}_{i,j},\hfill \end{array}$$ (69)

where ${\widehat{T}}_{i,j}=O\left(h_x^6\right)$.

At every $\left(x_i,y_j\right)$, let

$$B_{i,j}=2h_x^2\left[3{\widehat{\widehat{R}}}_{i,j}+{\widehat{R}}_{i+\frac{1}{2},j}+{\widehat{R}}_{i-\frac{1}{2},j}+{\widehat{R}}_{i,j+\frac{1}{2}}+{\widehat{R}}_{i,j-\frac{1}{2}}-4{\tilde{\tilde{R}}}_{i,j}\right]+BoundaryValues,$$

and $E_{i,j}=u_{i,j}-U_{i,j}$. Taking $S=B,E,u,U$ and$\widehat{T}$, we denote the corresponding column matrix as S, i.e.

$$\mathit{S}={\left[S_{1,1},S_{2,1},\dots ,S_{N_x-1,1},S_{1,2},S_{2,2},\dots ,S_{N_x-1,2},\dots ,S_{1,N_x-1},S_{2,N_x-1},\dots ,S_{N_x-1,N_x-1}\right]}_{{\left(N_x-1\right)}^2\times 1}^t,$$

t representing the transpose of the concerned matrix.

Then, the matrix form of Eq. (69) is given by:

$$\mathbf{Du}+\mathbf{B}\left(\mathbf{u}\right)=0,$$ (70)

where

$\mathit{D}={[\mathrm{A},\mathrm{C},\mathrm{A}]}_{{(N_x-1)}^2\times {(N_x-1)}^2}$ (Tri-block-diagonal matrix) for

$\mathit{C}={[{\gamma }^2\Psi -5\Phi ,10(\Phi +{\gamma }^2\Psi ),{\gamma }^2\Psi -5\Phi ]}_{(N_x-1)\times (N_x-1)}$ (Tri-diagonal matrix)

$\mathit{A}={[-\left(\frac{\Phi +{\gamma }^2\Psi }{2}\right),\Phi -5{\gamma }^2\Psi ,-\left(\frac{\Phi +{\gamma }^2\Psi }{2}\right)]}_{(N_x-1)\times (N_x-1)}$ (Tri-diagonal matrix) with the following assumptions:

$${\gamma }^2\Psi -5\Phi >0,$$ (71)

$$\Phi -5{\gamma }^2\Psi >0,$$ (72)

which yield $\Phi >0$ and $\Psi >0$. Hence, D has all positive diagonal elements and all negative off-diagonal elements.

Also, we know that

$$\mathit{DU}+\mathit{B}\left(\mathit{U}\right)+\widehat{\mathit{T}}=0,$$ (73)

U being the exact solution vector, and ${\widehat{T}}_{i,j}=O\left(h_x^6\right)$.

Now, let

$${\widehat{r}}_{i\pm \frac{1}{2},j}=R\left(x_{i\pm \frac{1}{2}},y_j,{\widehat{u}}_{i\pm \frac{1}{2},j},{\widehat{u}}_{xi\pm \frac{1}{2},j},{\widehat{u}}_{yi\pm \frac{1}{2},j}\right)\cong {\widehat{R}}_{i\pm \frac{1}{2},j},$$

$${\widehat{r}}_{i,j\pm \frac{1}{2}}=R\left(x_i,y_{j\pm \frac{1}{2}},{\widehat{u}}_{i,j\pm \frac{1}{2}},{\widehat{u}}_{xi,j\pm \frac{1}{2}},{\widehat{u}}_{yi,j\pm \frac{1}{2}}\right)\cong {\widehat{R}}_{i,j\pm \frac{1}{2}},$$

$${\widehat{\widehat{r}}}_{i,j}=R\left(x_i,y_j,u_{i,j},{\widehat{\widehat{u}}}_{xi,j},{\widehat{\widehat{u}}}_{yi,j}\right)\cong {\widehat{\widehat{R}}}_{i,j},$$

$${\tilde{\tilde{r}}}_{i,j}=R\left(x_i,y_j,{\tilde{\tilde{u}}}_{i,j},{\tilde{\tilde{u}}}_{xi,j},{\tilde{\tilde{u}}}_{yi,j}\right)\cong {\tilde{\tilde{R}}}_{i,j}.$$

Then, it is easy to obtain the following:

$${\widehat{r}}_{i\pm \frac{1}{2},j}-{\widehat{R}}_{i\pm \frac{1}{2},j}=\left({\widehat{u}}_{i\pm \frac{1}{2},j}-{\widehat{U}}_{i\pm \frac{1}{2},j}\right)G_{i\pm \frac{1}{2},j}^{\left(1\right)}+\left({\widehat{u}}_{xi\pm \frac{1}{2},j}-{\widehat{U}}_{xi\pm \frac{1}{2},j}\right)H_{i\pm \frac{1}{2},j}^{\left(1\right)}+\left({\widehat{u}}_{yi\pm \frac{1}{2},j}-{\widehat{U}}_{yi\pm \frac{1}{2},j}\right)I_{i\pm \frac{1}{2},j}^{\left(1\right)},$$ (74)

$${\widehat{r}}_{i,j\pm \frac{1}{2}}-{\widehat{R}}_{i,j\pm \frac{1}{2}}=\left({\widehat{u}}_{i,j\pm \frac{1}{2}}-{\widehat{U}}_{i,j\pm \frac{1}{2}}\right)G_{i,j\pm \frac{1}{2}}^{\left(2\right)}+\left({\widehat{u}}_{xi,j\pm \frac{1}{2}}-{\widehat{U}}_{xi,j\pm \frac{1}{2}}\right)H_{i,j\pm \frac{1}{2}}^{\left(2\right)}++\left({\widehat{u}}_{yi,j\pm \frac{1}{2}}-{\widehat{U}}_{yi,j\pm \frac{1}{2}}\right)I_{i,j\pm \frac{1}{2}}^{\left(2\right)},$$ (75)

$${\widehat{\widehat{r}}}_{i,j}-{\widehat{\widehat{R}}}_{i,j}=\left({\widehat{\widehat{u}}}_{xi,j}-{\widehat{\widehat{U}}}_{xi,j}\right)H_{i,j}^{\left(3\right)}+\left({\widehat{\widehat{u}}}_{yi,j}-{\widehat{\widehat{U}}}_{yi,j}\right)I_{i,j}^{\left(3\right)},$$ (76)

$${\tilde{\tilde{r}}}_{i,j}-{\tilde{\tilde{R}}}_{i,j}=\left({\tilde{\tilde{u}}}_{i,j}-{\tilde{\tilde{U}}}_{i,j}\right)G_{i,j}^{\left(3\right)}+\left({\tilde{\tilde{u}}}_{xi,j}-{\tilde{\tilde{U}}}_{xi,j}\right)H_{i,j}^{\left(3\right)}+\left({\tilde{\tilde{u}}}_{yi,j}-{\tilde{\tilde{U}}}_{yi,j}\right)I_{i,j}^{\left(3\right)},$$ (77)

for appropriate values of $Q_{i\pm \frac{1}{2},j}^{\left(1\right)}$, $Q_{i,j\pm \frac{1}{2}}^{\left(2\right)}$ and $Q_{i,j}^{\left(3\right)}$, where $Q=G$, $H$ and $I.$

Further, for $Q=H$ and I, we observe that

$$Q_{i\pm \frac{1}{2},j}^{\left(1\right)}=Q_{i,j}^{\left(1\right)}\pm \frac{h_x}{2}{\left(\frac{\partial Q^{\left(1\right)}}{\partial x}\right)}_{i,j}+O\left(h_x^2\right),$$ (78)

$$Q_{i,j\pm \frac{1}{2}}^{\left(2\right)}=Q_{i,j}^{\left(2\right)}\pm \frac{h_x}{2\gamma }{\left(\frac{\partial Q^{\left(2\right)}}{\partial y}\right)}_{i,j}+O\left(h_x^2\right)$$ (79)

and

$$G_{i\pm \frac{1}{2},j}^{\left(1\right)}=G_{i,j}^{\left(1\right)}\pm O\left(h_x\right),$$ (80)

$$G_{i,j\pm \frac{1}{2}}^{\left(2\right)}=G_{i,j}^{\left(2\right)}\pm O\left(h_x\right).$$ (81)

Thus, using the Eqs. (74)–(81), we get

$$\mathit{B}\left(\mathit{u}\right)-\mathit{B}\left(\mathit{U}\right)=\mathit{FE},$$ (82)

where $\mathrm{F}=\left({\mathit{F}}_{\zeta ,\eta }\right)$, [$\zeta ,\eta =1,2,\dots ,{\left(N_x-1\right)}^2]$ is the block tri-diagonal matrix such that:

$$\begin{array}{ccc}& & F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-1\right)\left(N_x-1\right)+i}\hfill \\ & & =h_x^2[2G_{i,j}^{\left(1\right)}+2G_{i,j}^{\left(2\right)}-6G_{i,j}^{\left(3\right)}+\frac{1}{\Phi }{H}_{i,j}^{\left(1\right)}{H}_{i,j}^{\left(3\right)}+\frac{1}{\Psi }{I}_{i,j}^{\left(2\right)}{I}_{i,j}^{\left(3\right)}-2{\left(\frac{\partial H^{\left(1\right)}}{\partial x}\right)}_{i,j}-2\gamma {\left(\frac{\partial I^{\left(2\right)}}{\partial y}\right)}_{i,j}]+O\left(h_x^4\right);i,j\in \left[1,N_x-1\right]\cap \mathbb{Z},\hfill \end{array}$$

$$\begin{array}{ccc}& & F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-1\right)\left(N_x-1\right)+i\pm 1}\hfill \\ & & =h_x\left[\pm 2H_{i,j}^{\left(1\right)}\pm H_{i,j}^{\left(2\right)}\mp \frac{1}{2}\left(1+\frac{\Psi {\gamma }^2}{\Phi }\right)H_{i,j}^{\left(3\right)}\right]\hfill \\ & & +\frac{h_x^2}{2}\left[2G_{i,j}^{\left(1\right)}+2{\left(\frac{\partial H^{\left(1\right)}}{\partial x}\right)}_{i,j}-G_{i,j}^{\left(3\right)}-\frac{1}{\Phi }{H}_{i,j}^{\left(1\right)}{H}_{i,j}^{\left(3\right)}\right]+O\left(h_x^3\right);\hfill \\ & & \left\{i\in \left[1,N_x-2\right]\cap \mathbb{Z},j\in \left[1,N_x-1\right]\cap \mathbb{Z}\right\},\left\{i\in \left[2,N_x-1\right]\cap \mathbb{Z},j\in \left[1,N_x-1\right]\cap \mathbb{Z}\right\},\hfill \end{array}$$

$$\begin{array}{ccc}& & F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-1\pm 1\right)\left(N_x-1\right)+i}=\hfill \\ & & =\frac{h_x}{2}\left[\pm 2\gamma I_{i,j}^{\left(1\right)}\pm 4\gamma I_{i,j}^{\left(2\right)}\mp \left(\gamma +\frac{\Phi }{\Psi \gamma }\right)I_{i,j}^{\left(3\right)}\right]+\frac{h_x^2}{2}\left[2G_{i,j}^{\left(2\right)}-G_{i,j}^{\left(3\right)}+2\gamma {\left(\frac{\partial I^{\left(2\right)}}{\partial y}\right)}_{i,j}-\frac{1}{\Psi }{I}_{i,j}^{\left(2\right)}{I}_{i,j}^{\left(3\right)}\right]\hfill \\ & & +O\left(h_x^3\right);\left\{i\in \left[1,N_x-1\right]\cap \mathbb{Z},j\in \left[1,N_x-2\right]\cap \mathbb{Z}\right\},\left\{i\in \left[1,N_x-1\right]\cap \mathbb{Z},j\in \left[2,N_x-1\right]\cap \mathbb{Z}\right\},\hfill \end{array}$$

$$\begin{array}{ccc}& & F_{\left(j-1\right)\left(N_x-1\right)+i,j\left(N_x-1\right)+i\pm 1}\hfill \\ & & =\frac{h_x}{4}\left[2\gamma I_{i,j}^{\left(1\right)}\pm 2H_{i,j}^{\left(2\right)}\pm \left(\frac{\Psi {\gamma }^2}{\Phi }-1\right)H_{i,j}^{\left(3\right)}+\left(-\gamma +\frac{\Phi }{\Psi \gamma }\right)I_{i,j}^{\left(3\right)}\right]\hfill \\ & & +\frac{h_x^2}{8}\left[\pm 2\gamma {\left(\frac{\partial I^{\left(1\right)}}{\partial x}\right)}_{i,j}\pm 2{\left(\frac{\partial H^{\left(2\right)}}{\partial y}\right)}_{i,j}\mp \frac{\gamma }{\Phi }{I}_{i,j}^{\left(1\right)}{H}_{i,j}^{\left(3\right)}\mp \frac{1}{\Psi \gamma }{H}_{i,j}^{\left(2\right)}{I}_{i,j}^{\left(3\right)}\right]\hfill \\ & & +O\left(h_x^3\right);\left\{i\in \left[1,N_x-2\right]\cap \mathbb{Z},j\in \left[1,N_x-2\right]\cap \mathbb{Z}\right\},\left\{i\in \left[2,N_x-1\right]\cap \mathbb{Z},j\in \left[1,N_x-2\right]\cap \mathbb{Z}\right\},\hfill \end{array}$$

$$\begin{array}{ccc}& & F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-2\right)\left(N_x-1\right)+i\pm 1}\hfill \\ & & =\frac{h_x}{4}\left[-2\gamma I_{i,j}^{\left(1\right)}\pm 2H_{i,j}^{\left(2\right)}\pm \left(\frac{\Psi {\gamma }^2}{\Phi }-1\right)H_{i,j}^{\left(3\right)}+\left(\gamma -\frac{\Phi }{\gamma \Psi }\right)I_{i,j}^{\left(3\right)}\right]\hfill \\ & & +\frac{h_x^2}{8}\left[\mp 2\gamma {\left(\frac{\partial I^{\left(1\right)}}{\partial x}\right)}_{i,j}\mp 2{\left(\frac{\partial H^{\left(2\right)}}{\partial y}\right)}_{i,j}\pm \frac{\gamma }{\Phi }{I}_{i,j}^{\left(1\right)}{H}_{i,j}^{\left(3\right)}\pm \frac{1}{\Psi \gamma }{H}_{i,j}^{\left(2\right)}{I}_{i,j}^{\left(3\right)}\right]\hfill \\ & & +O\left(h_x^3\right);\left\{i\in \left[1,N_x-2\right]\cap \mathbb{Z},j\in \left[2,N_x-1\right]\cap \mathbb{Z}\right\},\hfill \\ & & \left\{i\in \left[2,N_x-1\right]\cap \mathbb{Z},j\in \left[2,N_x-1\right]\cap \mathbb{Z}\right\},\hfill \end{array}$$

Using Eqs. (70) and (73), with the aid of (82), we obtain

$$\left(\mathit{D}+\mathit{F}\right)\mathit{E}=\widehat{\mathit{T},}$$ (83)

which is called the error equation.

Assuming $\overline{\Omega }=\Omega \cup \partial \Omega$, let

$G_{*}=$Min $\left(\frac{\partial R}{\partial u}\right)$ on $\overline{\Omega }$ and $G^{*}=$Max $\left(\frac{\partial R}{\partial u}\right)$ on $\overline{\Omega }$.

Then, clearly

$$0<G_{*}\le G_{i\pm \frac{1}{2},j}^{\left(1\right)},G_{i,j\pm \frac{1}{2}}^{\left(2\right)},G_{i,j}^{\left(3\right)}\le G^{*}.$$

Further, for $Q=H$ and $I$, let

$$0<\left|Q_{i\pm \frac{1}{2},j}^{\left(1\right)}\right|,\left|Q_{i,j\pm \frac{1}{2}}^{\left(2\right)}\right|,\left|Q_{i,j}^{\left(3\right)}\right|\le Q$$

and

$$\left|{\left(\frac{\partial Q^{\left(1\right)}}{\partial x}\right)}_{i,j}\right|\le Q^{\left(1\right)}\text{and}\left|{\left(\frac{\partial Q^{\left(2\right)}}{\partial y}\right)}_{i,j}\right|\le Q^{\left(2\right)},$$

$Q$, $Q^{\left(1\right)}$, $Q^{\left(2\right)}>0$ being constants.

Hence, for appropriately small values of $h_x$,

$$\left|F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-1\right)\left(N_x-1\right)+i}\right|<10\left(\Phi +{\gamma }^2\Psi \right);i,j\in \left[1,N_x-1\right]\cap \mathbb{Z},$$

$$\left|F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-1\right)\left(N_x-1\right)+i\pm 1}\right|<{\gamma }^2\Psi -5\Phi ,\left\{i\in \left[1,N_x-2\right]\cap \mathbb{Z},j\in \left[1,N_x-1\right]\cap \mathbb{Z}\right\},\left\{i\in \left[2,N_x-1\right]\cap \mathbb{Z},j\in \left[1,N_x-1\right]\cap \mathbb{Z}\right\},$$

$$\left|F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-1\pm 1\right)\left(N_x-1\right)+i}\right|<{\gamma }^2\Psi -5\Phi ,\left\{i\in \left[1,N_x-1\right]\cap \mathbb{Z},j\in \left[1,N_x-2\right]\cap \mathbb{Z}\right\},\left\{i\in \left[1,N_x-1\right]\cap \mathbb{Z},j\in \left[2,N_x-1\right]\cap \mathbb{Z}\right\},$$

$$\left|F_{\left(j-1\right)\left(N_x-1\right)+i,j\left(N_x-1\right)+i\pm 1}\right|<-\left(\frac{\Phi +{\gamma }^2\Psi }{2}\right),\left\{i\in \left[1,N_x-2\right]\cap \mathbb{Z},j\in \left[1,N_x-2\right]\cap \mathbb{Z}\right\},\left\{i\in \left[2,N_x-1\right]\cap \mathbb{Z},j\in \left[1,N_x-2\right]\cap \mathbb{Z}\right\},$$

$$\begin{array}{ccc}& & \left|F_{\left(j-1\right)\left(N_x-1\right)+i,\left(j-2\right)\left(N_x-1\right)+i\pm 1}\right|<-\left(\frac{\Phi +{\gamma }^2\Psi }{2}\right);\left\{i\in \left[1,N_x-2\right]\cap \mathbb{Z},j\in \left[2,N_x-1\right]\cap \mathbb{Z}\right\},\hfill \\ & & \left\{i\in \left[2,N_x-1\right]\cap \mathbb{Z},j\in \left[2,N_x-1\right]\cap \mathbb{Z}\right\}.\hfill \end{array}$$

It can be seen $\mathit{D}+\mathit{F}$ is an irreducible matrix, since it has strongly connected directed graph (see Varga [3]).

Now, let us assume that $S_l$ denotes the sum of the $l^{th}$ row entries of the matrix $\mathit{D}+\mathit{F}$. It is easy to obtain the following:

For $l=1,N_x-1:$

$$S_l=\frac{11\left({\gamma }^2\Psi +\Phi \right)}{2}+h_x^2\left(3G_{l,1}^{\left(1\right)}+3G_{l,1}^{\left(2\right)}-7G_{l,1}^{\left(3\right)}\right)+\frac{h_x}{8}\left(b_l+h_x{c}_l\right)+O\left(h_x^3\right),$$ (84)

where

$$b_l=\pm 16H_{l,1}^{\left(1\right)}\pm 12H_{l,1}^{\left(2\right)}\mp \left(6+\frac{2\Psi {\gamma }^2}{\Phi }\right)H_{l,1}^{\left(3\right)}+12\gamma I_{l,1}^{\left(1\right)}+16\gamma I_{l,1}^{\left(2\right)}-2\left(3\gamma +\frac{\Phi }{\Psi \gamma }\right)I_{l,1}^{\left(3\right)},$$

$$c_l=\frac{4}{\Phi }{H}_{l,1}^{\left(1\right)}{H}_{l,1}^{\left(3\right)}+\frac{4}{\Psi }{I}_{l,1}^{\left(2\right)}{I}_{l,1}^{\left(3\right)}-8{\left(\frac{\partial H^{\left(1\right)}}{\partial x}\right)}_{l,1}-8\gamma {\left(\frac{\partial I^{\left(2\right)}}{\partial y}\right)}_{l,1}\mp \frac{\gamma }{\Phi }{I}_{l,1}^{\left(1\right)}{H}_{l,1}^{\left(3\right)}\mp \frac{1}{\Psi \gamma }{I}_{l,1}^{\left(3\right)}{H}_{l,1}^{\left(2\right)\pm 2\gamma {\left(\frac{\partial I^{\left(1\right)}}{\partial x}\right)}_{l,1}\pm 2{\left(\frac{\partial H^{\left(2\right)}}{\partial y}\right)}_{l,1}.}$$

$$S_{\left(N_x-1\right)\left(N_x-2\right)+l}=\frac{11\left({\gamma }^2\Psi +\Phi \right)}{2}+\frac{h_x^2}{2}\left(3G_{l,N_x-1}^{\left(3\right)}+3G_{l,N_x-1}^{\left(2\right)}-7G_{l,N_x-1}^{\left(3\right)}\right)+\frac{h_x}{8}\left(b_l+h_x{c}_l\right)+O\left(h_x^3\right),$$ (85)

where

$$b_l=\pm 16H_{l,N_x-1}^{\left(1\right)}\pm 12H_{l,N_x-1}^{\left(2\right)}\mp 2\left(3+\frac{\Psi {\gamma }^2}{\Phi }\right)H_{l,N_x-1}^{\left(3\right)}-12\gamma I_{l,N_x-1}^{\left(1\right)}-16\gamma I_{l,N_x-1}^{\left(1\right)}-16\gamma I_{l,N_x-1}^{\left(2\right)}+\left(3\gamma +\frac{\Phi }{\Psi \gamma }\right)I_{l,N_x-1}^{\left(3\right)},$$

$$\begin{array}{ccc}\hfill c_l& =& \frac{4}{\Phi }{H}_{l,N_x-1}^{\left(1\right)}{H}_{l,N_x-1}^{\left(3\right)}+\frac{4}{\Psi }{I}_{l,N_x-1}^{\left(2\right)}{I}_{l,N-1}^{\left(3\right)}\pm \frac{\gamma }{\Phi }{I}_{l,N_x-1}^{\left(1\right)}{H}_{k,N_x-1}^{\left(3\right)}\hfill \\ & & \pm \frac{1}{\Psi \gamma }{I}_{l,N_x-1}^{\left(3\right)}{H}_{l,N_x-1}^{\left(2\right)}-8{\left(\frac{\partial H^{\left(1\right)}}{\partial x}\right)}_{l,N_x-1}-8\gamma {\left(\frac{\partial I^{\left(2\right)}}{\partial y}\right)}_{l,N_x-1}\mp 2\gamma {\left(\frac{\partial I^{\left(1\right)}}{\partial x}\right)}_{l,N_x-1}\mp 2{\left(\frac{\partial H^{\left(2\right)}}{\partial y}\right)}_{l,N_x-1}.\hfill \end{array}$$

For $q\in \left[2,N_x-2\right]\cap \mathbb{Z}$

$$S_{\left(q-1\right)\left(N_x-1\right)+l}=6\Phi +\frac{h_x^2}{2}\left[6G_{l,q}^{\left(1\right)}+8G_{l,q}^{\left(2\right)}-15G_{l,q}^{\left(3\right)}\right]+\frac{h_x}{2}[b_{\left(q-1\right)\left(N_x-1\right)+l}+h_x{c}_{\left(q-1\right)\left(N_x-1\right)+l}]+O\left(h_x^3\right),$$ (86)

where

$$b_{\left(q-1\right)\left(N_x-1\right)+l}=\pm 4H_{l,q}^{\left(2\right)}\pm 4H_{l,q}^{\left(1\right)}\mp 2H_{l,q}^{\left(3\right)},$$

$$c_{\left(q-1\right)\left(N_x-1\right)+l}=-2{\left(\frac{\partial H^{\left(1\right)}}{\partial x}\right)}_{l,q}+\frac{1}{\Phi }{H}_{l,q}^{\left(1\right)}{H}_{l,q}^{\left(3\right)}.$$

For $r\in \left[2,N_x-2\right]\cap \mathbb{Z}$

$$S_{\left(l-1\right)\left(N_x-1\right)+r}=6{\gamma }^2\Psi +\frac{h_x^2}{2}\left[8G_{r,l}^{\left(1\right)}+6G_{r,l}^{\left(2\right)}-15G_{r,l}^{\left(3\right)}\right]+\frac{h_x}{2}\left[b_{\left(l-1\right)\left(N_x-1\right)+r}+h_x{c}_{\left(l-1\right)\left(N_x-1\right)+r}\right]++O\left(h_x^3\right),$$ (87)

where

$$b_{\left(l-1\right)\left(N_x-1\right)+r}=\pm 4\gamma I_{r,l}^{\left(1\right)}\pm 4\gamma I_{r,l}^{\left(2\right)}\mp 2\gamma I_{r,l}^{\left(3\right)},$$

$$c_{\left(l-1\right)\left(N_x-1\right)+r}=-2\gamma {\left(\frac{\partial I^{\left(2\right)}}{\partial y}\right)}_{r,l}+\frac{1}{\Psi }{I}_{r,l}^{\left(2\right)}{I}_{r,l}^{\left(3\right)}.$$

Also, for $q,r\in \left[2,N_x-2\right]\cap \mathbb{Z}$

$$S_{\left(r-1\right)\left(N_x-1\right)+q}=4h_x^2\left[G_{q,r}^{\left(1\right)}+G_{q,r}^{\left(2\right)}-2G_{q,r}^{\left(2\right)}\right]+O\left(h_x^4\right).$$ (88)

Using the Eqs. (85)–(88), we obtain

$$\left|b_l\right|\le \left(34+\frac{2\Psi {\gamma }^2}{\Phi }\right)H+\left(34\gamma +\frac{2\Phi }{\Psi \gamma }\right)I,$$

$$\left|c_l\right|\le 4\frac{H^2}{\Phi }+4\frac{I^2}{\Psi }+\left(\frac{\gamma }{\Phi }+\frac{1}{\Psi \gamma }\right)HI+8\left[H^{\left(1\right)}+I^{\left(2\right)}\right]+2\left[\gamma I^{\left(1\right)}+H^{\left(2\right)}\right]$$

for $l=1,N_x-1,\left(N_x-1\right)\left(N_x-2\right)+1,{\left(N_x-1\right)}^2.$

$$\left|b_l\right|\le 10H,$$

$$\left|c_l\right|\le 2H+\frac{H^2}{\Phi }$$

for $l=\left(q-1\right)\left(N_x-1\right)+1$ and $l=q\left(N_x-1\right)$, such that $q\in \left[2,N_x-2\right]\cap \mathbb{Z}$.

$$\left|b_l\right|\le 10\gamma H,$$

$$\left|c_l\right|\le 2I+\frac{I^2}{\Psi }$$

for $l=r$ and $l=\left(N_x-1\right)\left(N_x-2\right)+r$, such that $r\in \left[2,N_x-2\right]\cap \mathbb{Z}$.

Thus, using appropriately small values of $h_x$,

$$S_l>C_1{h}_x^2,$$

where $C_1>0$ is a constant, and $l=1,N_x-1,\left(N_x-1\right)\left(N_x-2\right)+1,{\left(N_x-1\right)}^2$.

$${\mathrm{S}}_{\mathrm{l}}>{\mathrm{C}}_2{\mathrm{h}}_{\mathrm{x}}^2,$$

where constant $C_2>0$ is a constant, and $l=\left(q-1\right)\left(N_x-1\right)+1,q\left(N_x-1\right)$ such that

$$\mathrm{q}\in \left[2,{\mathrm{N}}_{\mathrm{x}}-2\right]\cap \mathbb{Z}.$$

$${\mathrm{S}}_{\mathrm{l}}>{\mathrm{C}}_3{\mathrm{h}}_{\mathrm{x}}^2,$$

where $C_3>0$ is a constant, and $l=r,\left(N_x-1\right)\left(N_x-2\right)+r$such that $r\in \left[2,N_x-2\right]\cap \mathbb{Z}$.

$${\mathrm{S}}_{\left(r-1\right)\left(N-2\right)+q}\ge {\mathrm{C}}_4{\mathrm{h}}_{\mathrm{x}}^2,$$

where $C_4>0$ is a constant, under the assumption that for each $q,r$, $C_{q,r}=G_{q,r}^{\left(1\right)}+G_{q,r}^{\left(2\right)}-2G_{q,r}^{\left(3\right)}>0,$ such that $q,r\in \left[2,N_x-2\right]\cap \mathbb{Z}$.

Hence, $\mathrm{D}+\mathrm{F}$ is monotone and its inverse exists (Henrici [58]). Assuming ${\left(\mathrm{D}+\mathrm{F}\right)}^{-1}={\mathrm{K}}^{-1},$ where $\mathrm{K}=\left({\mathrm{K}}_{i,j}\right),$ for $1\le i,j\le {\left(N_x-1\right)}^2.$

We know that $\sum _{m=1}^{{\left(N_x-1\right)}^2}{K}_{p,m}{S}_m=1,$ hence for $p=1,2,\dots ,{(N_x-1)}^2$ we obtain

$$K_{p,l}\le \frac{1}{S_l}<\frac{1}{C_1{h}_x^2},$$ (89)

for $l=1,N_x-1,\left(N_x-1\right)\left(N_x-2\right)+1,{\left(N_x-1\right)}^2.$

$$\sum _{q=2}^{N_x-2}{K}_{p,l}<\frac{2}{C_2{h}_x^2},$$ (90)

for $l=\left(q-1\right)\left(N_x-1\right)+1,q\left(N_x-1\right),q=1,2,\dots ,N_x-2.$

$$\sum _{r=2}^{N_x-2}{K}_{p,l}<\frac{2}{C_3{h}_x^2},$$ (91)

for $l=r,\left(N_x-1\right)\left(N_x-2\right)+r,r=1,2,\dots ,N_x-2.$

$$\sum _{q=2}^{N_x-2}\sum _{r=2}^{N_x-2}{K}_{p,l}\le \frac{1}{C_4{h}_x^2},$$ (92)

for $l=\left(r-1\right)\left(N_x-1\right)+q$and$q,r=1,2,\dots ,N_x-2.$

From Eq. (83), we obtain

$$\parallel \mathrm{E}\parallel \le \parallel \mathrm{K}\parallel \parallel \widehat{\mathrm{T}}\parallel ,$$ (93)

where

$$\begin{array}{ccc}\hfill \parallel \mathbf{K}\parallel & =& Ma{x}_{1\le m\le {\left(N_x-1\right)}^2}[\left(K_{m,1}+\sum _{q=2}^{N_x-2}{K}_{m,q}+K_{m,N_x-1}\right)+\left(\sum _{q=2}^{N_x-2}{K}_{m,\left(q-1\right)\left(N_x-1\right)+1}+\sum _{q=2}^{N_x-2}\sum _{r=2}^{N_x-2}{K}_{m,\left(q-1\right)\left(N_x-1\right)+r}+\sum _{q=2}^{N_x-2}{K}_{m,q\left(N_x-1\right)}\right)\hfill \\ & & +\left(K_{m,\left(N_x-1\right)\left(N_x-2\right)+1}+\sum _{q=2}^{N-2}{K}_{m,\left(N_x-1\right)\left(N_x-2\right)+q}+K_{m,{\left(N_x-1\right)}^2}\right)]\hfill \end{array}$$ (94)

Thus, using Eqs. (89)–(92) and (94) in (93), for suitably small values of $h_x$, we obtain

$$\parallel \mathbf{E}\parallel \le O\left(h_x^4\right).$$ (95)

Hence, $O\left(h_x^4\right)$ accuracy of the developed FDM over (68) is established.

Validation of the proposed FDM

We solve six benchmark problems of physical significance, whose closed-form solutions are known. The boundary values and the right-hand side function of independent variables can be obtained using the given closed-form solution. We use appropriate iteration methods to solve the resultant system of difference equations. The initial approximation 0 is considered for all problems.

Problem 1

(Poisson's equation in r-z plane)

$$u_{rr}+u_{zz}+\frac{1}{r}{u}_r=g\left(r,z\right),0<z<e,0<r<1.$$ (96)

The solution in closed-form is$u\left(r,z\right)=coshr.coshz.$ Eq. (95) solves for the potential field, the gravitational field or the pressure field in fluid dynamics. The Maximum Absolute Errors (MAEs) are reported in Table 1. Fig. 1a and b portray the closed-form and computational solutions, respectively, for $N_x=N_y=40$.

Table 1.

(Problem 1): The MAEs.

$N_x=N_y$	Proposed Method	Method [56]	Method [35]	Method [36]
20	8.1827(−09)	1.6787(−08)	6.8123(−04)	6.8577(−04)
40	5.1496(−10)	1.1201(−09)	1.7276(−04)	1.7844(−04)
80	3.2209(−11)	7.1160(−11)	4.3891(−05)	4.4756(−05)

Fig. 1.

a: N_x=N_y = 40, the analytical solution. b: N_x=N_y = 40, the approximate solution.

Problem 2

(Poisson's equation in r-$\theta$ plane)

$$u_{rr}+\frac{1}{r^2}{u}_{\theta \theta }+\frac{1}{r}{u}_r=g\left(r,\theta \right),0<r<1,0<\theta <\pi .$$ (97)

The closed-form solution is$u\left(r,\theta \right)=r^{2}cos\theta .$ In Table 2, the MAEs are charted. The exact and computational solutions are graphed in Fig. 2a and b, respectively, for $N_x=N_y$= 40.

Table 2.

(Problem 2): The MAEs.

$N_x=N_y$	Proposed Method	Method [56]	Method [35]	Method [36]
20	1.0180(−08)	7.1575(−08)	4.1614(−05)	4.2123(−05)
40	6.4137(−10)	4.4011(−09)	1.0162(−05)	1.1188(−05)
80	4.0585(−11)	2.7257(−10)	2.5098(−06)	2.5816(−06)

Fig. 2.

a: N_x=N_y = 40, the analytical solution. b: N_x=N_y = 40, the approximate solution.

Problem 3

(Burgers' equation in x-y plane)

$$\nu \left(u_{xx}+u_{yy}\right)=u\left(u_x+u_y\right)+g\left(x,y\right),0<x<\frac{1}{\sqrt{2}},0<y<1.$$ (98)

The exact solution in closed-form is $u\left(x,y\right)=e^x.\text{sin}\left(\pi y\right)$. The Eq. (97) represents a turbulent flow. The solution u depicts the speed of the fluid at each spatial coordinate (x,y), in a steady state. In Table 3, the MAEs are given. The closed-form and computational values are portrayed in Fig. 3a and b, respectively, for $\nu =0.01,N_x=N_y$= 40.

Table 3.

(Problem 3): The MAEs.

$N_x=N_y$	Proposed Method		Method [56]		Method [35], Method [36]
	$\nu =0.01$	$\nu =0.001$	$\nu =0.01$	$\nu =0.001$	$\nu =0.01,0.001$
20	1.1522(−06)	5.3437(−06)	5.3311(−06)	1.4276(−05)	Unstable
40	7.2051(−08)	3.3533(−07)	3.3019(−07)	8.9755(−07)	Unstable
80	4.5439(−09)	2.1190(−08)	2.0567(−08)	5.5923(−08)	Unstable

Fig. 3.

a: N_x=N_y = 40, $\nu$ =0.01, the analytical solution. b: N_x=N_y = 40, $\nu$ =0.01, the approximate solution.

Problem 4

(NS equations in rectangular form)

$$\frac{1}{{\mathrm{R}}_{\mathrm{e}}}\left({\mathrm{u}}_{\text{xx}}+{\mathrm{u}}_{\text{yy}}\right)=u{\mathrm{u}}_{\mathrm{x}}+v{\mathrm{u}}_{\mathrm{y}}+F\left(\mathrm{x},\mathrm{y}\right),0<x<1,0<y<\frac{1}{\sqrt{2}},$$ (99a)

$$\frac{1}{{\mathrm{R}}_{\mathrm{e}}}\left({\mathrm{v}}_{\text{xx}}+{\mathrm{v}}_{\text{yy}}\right)=u{\mathrm{v}}_{\mathrm{x}}+v{\mathrm{v}}_{\mathrm{y}}+G\left(\mathrm{x},\mathrm{y}\right),0<x<1,0<y<\frac{1}{\sqrt{2}}.$$ (99b)

The analytical solution of the above coupled equations is $u\left(x,y\right)=\sin \left(\pi x\right)\text{sin}\left(\pi y\right),$ $v\left(x,y\right)=\cos \left(\pi x\right)\text{cos}\left(\pi y\right).$ The solution u = (u, v) represents the velocity of the fluid at each spatial coordinate $\left(x,y\right)$, in a steady state. Table 4 shows the MAEs. The analytical and computational solutions for u and v are graphed in Fig. 4a–d for $N_x=N_y$= 40 and $R_e={10}^4.$

Table 4.

(Problem 4): The MAEs.

$N_x=N_y$		Proposed Method		Method [56]		Method [35], Method [36]
		$R_e={10}^2$	$R_e={10}^4$	$R_e={10}^2$	$R_e={10}^4$	$R_e={10}^2,{10}^4$
20	u	7.6329(−06)	2.2697(−03)	3.0187(−05)	2.7914(−03)	Unstable
	v	6.8159(−06)	1.7877(−03)	1.4415(−05)	2.4822(−03)
40	u	4.7752(−07)	1.4276(−04)	1.8831(−06)	1.7423(−04)	Unstable
	v	4.2861(−07)	1.1255(−04)	9.0549(−07)	1.6515(−04)
80	u	2.9987(−08)	8.9755(−06)	1.1785(−07)	1.0844(−05)	Unstable
	v	2.6949(−08)	7.1082(−06)	5.5772(−08)	1.0095(−05)

Fig. 4.

a: N_x=N_y = 40, R_e=10⁴, the analytical solution u. b: N_x=N_y = 40, R_e=10⁴, the numerical solution U. c: N_x=N_y = 40, R_e=10⁴, the analytical solution v. d: N_x=N_y = 40, R_e=10⁴, the numerical solution V.

Problem 5

(NS equations in cylindrical form)

$$\frac{1}{{\mathrm{R}}_{\mathrm{e}}}\left({\mathrm{u}}_{\text{rr}}+\frac{1}{\mathrm{r}}{\mathrm{u}}_{\mathrm{r}}+{\mathrm{u}}_{\text{zz}}-\frac{1}{{\mathrm{r}}^2}\mathrm{u}\right)=u{\mathrm{u}}_{\mathrm{r}}+v{\mathrm{u}}_{\mathrm{z}}+F\left(\mathrm{r},\mathrm{z}\right),0<z<\sqrt{2},0<r<1,$$ (100a)

$$\frac{1}{{\mathrm{R}}_{\mathrm{e}}}\left(v_{rr}+\frac{1}{r}{v}_r+v_{zz}\right)=u{v}_r+v{v}_z+G\left(r,z\right),0<z<\sqrt{2},0<r<1.$$ (100b)

The analytical solution of the above coupled equations is $u\left(r,z\right)=r^3\text{sinh}z,$ $v\left(r,z\right)=-4r^{2}coshz.$ The solution u = (u, v) represents the velocity of the fluid at each spatial coordinate $\left(r,z\right)$, in a steady state. Table 5 illustrates the MAEs. Fig. 5a–d represent the analytical and computational solutions obtained with $N_x=N_y$= 40 and $R_e={10}^3.$

Table 5.

(Problem 5): The MAEs.

$N_x=N_y$		Proposed Method		Method [56]		Method [35], Method [36]
		$R_e={10}^2$	$R_e={10}^3$	$R_e={10}^2$	$R_e={10}^3$	$R_e={10}^2,{10}^3$
20	u	7.5524(−06)	4.6032(−04)	2.9077(−05)	2.1550(−03)	Unstable
	v	4.7149(−06)	3.8555(−04)	1.6887(−05)	1.7179(−03)
40	u	4.6978(−07)	2.8856(−05)	1.7913(−06)	1.3304(−04)	Unstable
	v	2.9582(−07)	2.4389(−05)	1.1301(−06)	1.0721(−04)
80	u	2.9461(−08)	1.8165(−06)	1.1042(−07)	8.3305(−06)	Unstable
	v	1.8594(−08)	1.5378(−06)	7.0160(−08)	6.6892(−06)

Fig. 5.

a: N_x=N_y = 40, R_e=10³, the analytical solution u. b: N_x=N_y = 40, R_e=10³, the numerical solution U. c: N_x=N_y = 40, R_e=10³, the analytical solution v. d: N_x=N_y = 40, R_e=10³, the approximate solution V.

Problem 6

(Quasi-linear elliptic equation)

$${\mathrm{u}}_{\text{xx}}+\left(1+{\mathrm{u}}^2\right){\mathrm{u}}_{\text{yy}}=u\left({\mathrm{u}}_{\mathrm{x}}+{\mathrm{u}}_{\mathrm{y}}\right)+F\left(\mathrm{x},\mathrm{y}\right),0<x<e,0<y<\frac{\pi }{2}.$$ (101)

This is a general equation to depict the applicability of our methods over quasi-linear counterpart of BVPEs in concern. The analytical solution in closed form is $u\left(x,y\right)=e^x\text{siny}.$ In Table 6, the MAEs are shown. The closed-form and computational solutions are graphed in Fig. 6a and b with $N_x=N_y$= 40.

Table 6.

(Problem 6): The MAEs.

$N_x=N_y$	Proposed Method	Method [56]	Method [35]	Method [36]
20	6.6937(−06)	2.3722(−05)	2.3172(−04)	2.3868(−04)
40	4.2085(−07)	1.4792(−06)	5.8201(−05)	5.8816(−05)
80	2.6148(−08)	9.2983(−08)	1.4469(−05)	1.5275(−05)

Fig. 6.

a: N_x=N_y = 40, the analytical solution. b: N_x=N_y = 40, the numerical solution.

In summarization, the proposed stable higher order method provides tremendous practical advantage in terms of diminishing the required number of storages in comparison to lower order methods and also the overall computing time for a desired solution. The order of accuracy of the obtained solution is $O\left(h_x^4+h_x^2{h}_y^2+h_y^4\right)$. When $h_x$ is proportional to $h_y$, the order becomes either$O\left(h_x^4\right)$ or $O\left(h_y^4\right)$, since $h_x=\gamma h_y,\gamma \ne 1.$ The order of accuracy of the method developed in this article is $O\left(h_x^4+h_x^2{h}_y^2+h_y^4\right)$. When $h_x$ is proportional to $h_y$, the order becomes either$O\left(h_x^4\right)$ or $O\left(h_y^4\right)$, since $h_x=\gamma h_y,\gamma \ne 1.$ The rate of convergence is given by

$$\frac{\log \left(\frac{{\mathrm{E}}_{{\mathrm{h}}_{\mathrm{x}1}}}{{\mathrm{E}}_{\mathrm{h}{\mathrm{x}}_2}}\right)}{\log \left(\frac{{\mathrm{h}}_{\mathrm{x}1}}{{\mathrm{h}}_{\mathrm{x}2}}\right)},$$ (101)

$E_{h_{x1}}$ and $E_{h_{x2}}$ being the MAEs for two different mesh sizes $h_{x1}=\frac{x_{\alpha }}{N_{x1}}$and $h_{x2}=\frac{x_{\beta }}{N_{x2}}$, respectively. For calculation of rate of convergence, we have chosen, $N_{x1}=40$ and $N_{x2}=80.$ and the results are reported in Table 7.

Table 7.

(The rate of convergence): $h_{x1}=\frac{x_{\alpha }}{40},h_{x2}=\frac{x_{\alpha }}{80}.$

Problem	Involved parameters		Order
1	–		3.99
2	–		3.98
3	$\nu =0.01$		3.98
	$\nu =0.001$		3.98
4	Re=102	for u	3.99
		for v	3.99
	Re=104	for u	3.99
		for v	3.98
5	Re=102	for u	3.98
		for v	3.99
	Re=103	for u	3.99
		for v	3.98
6			4.00

The log-log error plots of the MAEs provided in the second column of Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 are represented in Fig. 7.

Fig. 7.

Order of Convergence: Problem 1 to Problem 6.

The computational results applied over some benchmark problems of high importance confirmed that stable solution near the singular point is generated even for high Reynolds number. It has been observed that the MAEs reduce by about $1/2^4$, when the grid size $h_x$ reduced to $h_x/2$. The fourth-order convergence of the suggested method is endorsed by the computational results. The graphs of the numerical and analytical solutions appear identical. The proposed method is useful to construct a new method in exponential form for 3D nonlinear BVPEs on an unequal mesh.

Data availability

No data was used for the research described in the article.

CRediT authorship contribution statement

Nikita Setia: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. R.K. Mohanty: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability

• No data was used for the research described in the article.