Error Analysis in the Numerical Solution of 3d Convection-Diffusion Equation by Finite Difference Methods

Romão, Estaner Claro; Silva, João Batista Campos; Moura, Luiz Felipe Mendes de

doi:10.5380/reterm.v8i1.61875

Cited by 4 publications

(3 citation statements)

References 3 publications

(8 reference statements)

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“…The physical quantities , c p , and ρ denote thermal conductivity, specific heat, and specific mass respectively for the conducting material. The velocity component in x-, y-, and z-directions is u, v, and w respectively [35,36]. Since the solution changes sharply with a change in the value of parameter τ , we analyze them for = 1 and different values of τ in the following cases.…”

Section: Table 1bmentioning

confidence: 99%

Exponential basis and exponential expanding grids third (fourth)-order compact schemes for nonlinear three-dimensional convection-diffusion-reaction equation

Jha

Singh

2019

Adv Differ Equ

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This paper addresses exponential basis and compact formulation for solving three-dimensional convection-diffusion-reaction equations that exhibit an accuracy of order three or four depending on exponential expanding or uniformly spaced grid network. The compact formulation is derived with three grid points in each spatial direction and results in a block-block tri-diagonal Jacobian matrix, which makes it more suitable for efficient computing. In each direction, there are two tuning parameters; one associated with exponential basis, known as the frequency parameter, and the other one is the grid ratio parameter that appears in exponential expanding grid sequences. The interplay of these parameters provides more accurate solution values in short computing time with less memory space, and their estimates are determined according to the location of layer concentration. The Jacobian iteration matrix of the proposed scheme is proved to be monotone and irreducible. Computational experiments with convection dominated diffusion equation, Schrödinger equation, Helmholtz equation, nonlinear elliptic Allen-Cahn equation, and sine-Gordon equation support the theoretical convergence analysis.

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Section: Table 1bmentioning

confidence: 99%

Exponential basis and exponential expanding grids third (fourth)-order compact schemes for nonlinear three-dimensional convection-diffusion-reaction equation

Jha

Singh

2019

Adv Differ Equ

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“…Ma and Ge [6] combined Richardson extrapolation and compact difference method of high-order for solving elliptic PDEs in three-dimensions using axial sweeping based on alternating direction implicit mechanism. Romão et al [7,8] described a least-square and Galerkin finite element methods to solve steady convection-diffusion-reaction. Gupta and Zhang [9] proposed a 19-point scheme discretization, parallelization, and vectorization.…”

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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“…Researcher Tamora James (2002) made a numerical solution of ADE for Radial Flow. Romao et al (2005) presented the finite difference methods of 3D convection diffusion equation to investigate error in the numerical solution of this equation. For this equation, Thongmoon and Mckibbin (2006) compared some numerical methods and indicated that FTCS and Crank-Nicolson scheme give…”

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confidence: 99%

Numerical solution of advection-diffusion equation using finite difference schemes

Andallah

Khatun

2020

Bangladesh J. Sci. Ind. Res.

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This paper presents numerical simulation of one-dimensional advection-diffusion equation. We study the analytical solution of advection diffusion equation as an initial value problem in infinite space and realize the qualitative behavior of the solution in terms of advection and diffusion co-efficient. We obtain the numerical solution of this equation by using explicit centered difference scheme and Crank-Nicolson scheme for prescribed initial and boundary data. We implement the numerical scheme by developing a computer programming code and present the stability analysis of Crank-Nicolson scheme for ADE. For the validity test, we perform error estimation of the numerical scheme and presented the numerical features of rate of convergence graphically. The qualitative behavior of the ADE for different choice of the advection and diffusion co-efficient is verified. Finally, we estimate the pollutant in a river at different times and different points by using these numerical scheme. Bangladesh J. Sci. Ind. Res.55(1), 15-22, 2020

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Error Analysis in the Numerical Solution of 3d Convection-Diffusion Equation by Finite Difference Methods

Cited by 4 publications

References 3 publications

Exponential basis and exponential expanding grids third (fourth)-order compact schemes for nonlinear three-dimensional convection-diffusion-reaction equation

Exponential basis and exponential expanding grids third (fourth)-order compact schemes for nonlinear three-dimensional convection-diffusion-reaction equation

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

Numerical solution of advection-diffusion equation using finite difference schemes

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