scholarly journals Two Dimensional Advection Diffusion Equations in Unstable Case

2021 ◽  
Vol 5 (1) ◽  
pp. 7
Author(s):  
Khaled Sadek Mohamed Essa ◽  
Sawsan Ibrahim Mohamed El Saied
Keyword(s):  
Diffusion Equations ◽  
Two Dimensional ◽  
Advection Diffusion ◽  
Unstable Case

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

A new spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations via Chebyshev polynomials

2019 ◽  
Vol 36 (7) ◽  
pp. 2327-2368 ◽  
Author(s):  
Mohsen Hadadian Nejad Yousefi ◽  
Seyed Hossein Ghoreishi Najafabadi ◽  
Emran Tohidi
Keyword(s):  
Integral Equation ◽  
Chebyshev Polynomials ◽  
Integral Equation Method ◽  
Diffusion Equations ◽  
Equation Method ◽  
Two Dimensional ◽  
Content Type ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Spectral Integral

Purpose The purpose of this paper is to develop an efficient and reliable spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations. Design/methodology/approach In this study, the considered two-dimensional unsteady advection-diffusion equations are transformed into the equivalent partial integro-differential equations via integrating from the considered unsteady advection-diffusion equation. After this stage, by using Chebyshev polynomials of the first kind and the operational matrix of integration, the integral equation would be transformed into the system of linear algebraic equations. Robustness and efficiency of the proposed method were illustrated by six numerical simulations experimentally. The numerical results confirm that the method is efficient, highly accurate, fast and stable for solving two-dimensional unsteady advection-diffusion equations. Findings The proposed method can solve the equations with discontinuity near the boundaries, the advection-dominated equations and the equations in irregular domains. One of the numerical test problems designed specially to evaluate the performance of the proposed method for discontinuity near boundaries. Originality/value This study extends the intention of one dimensional Chebyshev approximate approaches (Yuksel and Sezer, 2013; Yuksel et al., 2015) for two-dimensional unsteady advection-diffusion problems and the basic intention of our suggested method is quite different from the approaches for hyperbolic problems (Bulbul and Sezer, 2011).

FRACTIONAL STEP METHODS FOR SPECTRAL APPROXIMATION OF ADVECTION-DIFFUSION EQUATIONS

1996 ◽  
Vol 06 (07) ◽  
pp. 1027-1050 ◽  
Author(s):  
PAOLA GERVASIO
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Peclet Number ◽  
Diffusion Equations ◽  
Fractional Step ◽  
Spectral Approximation ◽  
Two Dimensional ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Large Peclet Number

Several classical fractional step schemes are proposed for the spectral approximation of advection-diffusion equation in two-dimensional geometries. Suitable boundary conditions are studied in order to preserve the accuracy of the schemes at each step. An example is given for the application of the fractional step schemes to solve problems with large Péclet number.

Three-Point Combined Compact Alternating Direction Implicit Difference Schemes for Two-Dimensional Time-Fractional Advection-Diffusion Equations

2015 ◽  
Vol 17 (2) ◽  
pp. 487-509 ◽  
Author(s):  
Guang-Hua Gao ◽  
Hai-Wei Sun
Keyword(s):  
Diffusion Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Order Accuracy ◽  
Alternating Direction Implicit ◽  
Truncation Errors ◽  
Advection Diffusion ◽  
Alternating Direction ◽  
Implicit Difference Schemes ◽  
Adi Schemes

AbstractThis paper is devoted to the discussion of numerical methods for solving two-dimensional time-fractional advection-diffusion equations. Two different three-point combined compact alternating direction implicit (CC-ADI) schemes are proposed and then, the original schemes for solving the two-dimensional problems are divided into two separate one-dimensional cases. Local truncation errors are analyzed and the unconditional stabilities of the obtained schemes are investigated by Fourier analysis method. Numerical experiments show the effectiveness and the spatial higher-order accuracy of the proposed methods.

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