A new high order compact off-step discretization for the system of 3D quasi-linear elliptic partial differential equations

Mohanty, R. K.; Setia, Nikita

doi:10.1016/j.apm.2013.02.018

Cited by 24 publications

(3 citation statements)

References 17 publications

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“…Example 4. Lastly, we solve a system of three nonlinear coupled elliptic PDEs modeling Navier-Stokes equations in Cartesian coordinates [21,40,44] 1 Re…”

Section: Numerical Illustrationsmentioning

confidence: 99%

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Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

Mkhatshwa

Khumalo

2022

Mathematics

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This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

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“…Example 4. Lastly, we solve a system of three nonlinear coupled elliptic PDEs modeling Navier-Stokes equations in Cartesian coordinates [21,40,44] 1 Re…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…Single PDEsExample 1. First, the method was tested on 3D Poisson's equation with singular behaviour[21,40]…”

mentioning

confidence: 99%

Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

Mkhatshwa

Khumalo

2022

Mathematics

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“…Wang et al [15] described a preconditioned conjugate gradient formula and applied it to solve general mesh step sizes compact scheme of fourth-order originated from the consistent discretization to the three-space Poisson's equation. An arithmetic average 19-point compact formulation to a system of quasilinear elliptic PDEs is obtained by Mohanty and Setia [16]. Gavete et al [17] considered the meshfree method as a generalization to finite difference discretization on irregular clouds of grids for computing the 3D parabolic and elliptic PDEs.…”

Section: Introductionmentioning

confidence: 99%

Fourth‐order compact scheme based on quasi‐variable mesh for three‐dimensional mildly nonlinear stationary convection–diffusion equations

Jha

Singh

2020

Numerical Methods Partial

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A new family of compact schemes of increased accuracy using quasi-variable mesh is presented for determining approximate solutions to the three-space dimensions mildly nonlinear convection dominated diffusion equations. The main thought behind the proposed scheme is to get uniformly distributed local truncation error, which otherwise not possible in case of finite-difference discretization using constant step-sizes mesh points. According to the zero or nonzero values of mesh stretching quantities, the increased accuracy fourth-order method refers to uniform meshes or quasi-variable meshes schemes. It is easy to tune the mesh points depending upon the location of subdomains having comparatively more fluctuating solution behavior. The block-tridiagonal construction of the Jacobian (iteration matrix) acquired with the new difference scheme makes it easier to implement with a reasonable computing time. We will describe the matrix and graph theoretic approach to analyze the convergence property and error estimate of the generalized scheme. A measure of accuracy, such as maximum errors, root-mean-squared errors, and numerical convergence rate, is examined by solving various forms of convection-dominated diffusion equations.

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A new high-accuracy method based on off-step cubic polynomial approximations for the solution of coupled Burgers’ equations and Burgers–Huxley equation

Mohanty

Sharma

2020

Engineering with Computers

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A new high order compact off-step discretization for the system of 3D quasi-linear elliptic partial differential equations

Cited by 24 publications

References 17 publications

Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

Fourth‐order compact scheme based on quasi‐variable mesh for three‐dimensional mildly nonlinear stationary convection–diffusion equations

A new high-accuracy method based on off-step cubic polynomial approximations for the solution of coupled Burgers’ equations and Burgers–Huxley equation

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