On The Existence and Asymptotic Stability of Periodic Contrast Structures in Quasilinear Reaction-Advection-Diffusion Equations

The k-epsilon turbulence model for incompressible flow involves two advection–diffusion equations plus point-source terms. We propose a new method for positivity analysis. This method uses an iterative procedure combined with an operator splitting. With this method we recover the well-known positivity result for the standard high Reynolds number model. Most importantly, we are able to prove the positivity result for general low Reynolds number k-epsilon models.

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A space-time backward substitution method for three-dimensional advection-diffusion equations

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2021.05.025 ◽

2021 ◽

Vol 97 ◽

pp. 77-85

Author(s):

HongGuang Sun ◽

Yi Xu ◽

Ji Lin ◽

Yuhui Zhang

Keyword(s):

Three Dimensional ◽

Space Time ◽

Diffusion Equations ◽

Substitution Method ◽

Advection Diffusion

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

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-12-2017-0511 ◽

2018 ◽

Vol 28 (11) ◽

pp. 2620-2649 ◽

Cited By ~ 2

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.

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