A new compact high order off-step discretization for the system of 2D quasi-linear elliptic partial differential equations

Abstract

A new fourth-order difference method for solving the system of two-dimensional quasi-linear elliptic equations is proposed. The difference scheme referred to as off-step discretization is applicable directly to the singular problems and problems in polar coordinates. Also, new fourth-order methods for obtaining the first-order normal derivatives of the solution are developed. The convergence analysis of the proposed method is discussed in details. The methods are applied to many physical problems to illustrate their accuracy and efficiency. MSC: 65N06

High precision compact numerical approximation in exponential form for the system of 2D quasilinear elliptic BVPs on a discrete irrational region

“…In 2021, Milewski [35] have proposed a meshless technique for solving a 2D linear EPDE with Dirichlet and Neumann conditions. Apart from the meshless approaches, Mohanty and Setia [36] have designed a highly accurate compact half-step discretization for a system of general non-linear 2D EPDEs in the year 2013. A compact FDM of high order accuracy for a 2D Poisson equation has been given by Zhai et al [37] in the following year.…”

“…The technique discussed in [49] can be used to obtain numerical approximation of order four for the quasilinear EPDE (1.1) . Further, the proposed method (6) for the single equation can be stretched out to the system of quasilinear EPDE in vector-matrix form, as discussed in [36] .…”

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

“… [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] .…”

“…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 approximation for BVPEs in vector-matrix system (see [33] ).…”

Operator compact exponential approximation for the solution of the system of 2D second order quasilinear elliptic partial differential equations

“…The following year Ananthakrishnaiah, Saldanha [17] discussed a fourth order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations. Thereafter, Mohanty et al [18][19][20][21], Zhang [22], Saldanha [23] discussed order h 4 difference methods for a class of elliptic boundary value problem. Dehghan et al [24] proposed preconditioning techniques to obtain faster convergence of the higher order methods applied to linear elliptic PDEs.…”