scholarly journals A finite Volume Runge-Kutta Ellam Method for the Solution of Advection Diffusion Equations

2001 ◽  
Vol 6 (2) ◽  
pp. 67 ◽  
Author(s):  
Mohammed Al-Lawatia ◽  
Robert C. Sharpley ◽  
Hong Wang
Keyword(s):  
Contaminant Transport ◽  
Finite Volume ◽  
Large Time ◽  
Numerical Solutions ◽  
Standard Test ◽  
Diffusion Equations ◽  
Governing Equations ◽  
Adjoint Methods ◽  
Advection Diffusion ◽  
Runge Kutta

We develop a finite volume characteristic method for the solution of the advection-diffusion equations which model the contaminant transport through porous medium. This method uses a second order Runge-Kutta approximation for the characteristics within the framework of the Eulerian Lagrangian localized adjoint methods (ELLAM). The derived scheme conserves mass, symmetrizes the governing equations and generates accurate numerical solutions even if large time steps are used.  Numerical experiments comparing several competitive methods using a standard test example are presented to illustrate the performance of the method.  

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.

scholarly journals Large Time Behavior of Nonlinear Finite Volume Schemes for Convection-Diffusion Equations

2020 ◽  
Vol 58 (5) ◽  
pp. 2544-2571
Author(s):  
Clément Cancès ◽  
Claire Chainais-Hillairet ◽  
Maxime Herda ◽  
Stella Krell
Keyword(s):  
Finite Volume ◽  
Large Time ◽  
Large Time Behavior ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Finite Volume Schemes ◽  
Convection Diffusion

scholarly journals General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
José A. Barrionuevo ◽  
Lucas S. Oliveira ◽  
Paulo R. Zingano
Keyword(s):  
Initial Data ◽  
Large Time ◽  
Heterogeneous Media ◽  
Diffusion Equations ◽  
Open Problems ◽  
New Techniques ◽  
One Dimensional ◽  
Advection Diffusion

We derive general bounds for the large time size of supnorm values ∥u(·,t)∥L∞(ℝ) of solutions to one-dimensional advection-diffusion equations ut+(b(x,t)u)x=uxx,x∈ℝ,t>0 with initial data u(·,0)∈Lp0(ℝ)∩L∞(ℝ) for some 1≤p0<∞ and arbitrary bounded advection speeds b(x,t), introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.

An Application of Discontinuous Galerkin Method for Blast Wave Simulations

2010 ◽  
Author(s):  
Emre Alpman
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Blast Wave ◽  
Finite Volume ◽  
Discontinuous Galerkin Method ◽  
Numerical Solutions ◽  
Blast Waves ◽  
Strong Discontinuities ◽  
Runge Kutta ◽  
Volume Method

An implementation of Runge-Kutta Discontinuous Galerkin method to an in-house computational fluid dynamics code capable of simulating blast waves was performed. The resultant code was tested for two shock tube problems with moderately and extremely strong discontinuities. Numerical solutions were compared with predictions of a finite volume method and exact solutions. It was observed that when there are extreme discontinuities in the flowfield, as in the case of blast waves, the limiter adopted for solution clearly affects the overall quality of the predictions. An alternative limiting technique was proposed and tested to improve the results obtained. Blast wave predictions using Runge-Kutta Discontinuous Galerkin method with the alternative limiting technique yielded slightly stronger and faster moving shock waves compared to finite volume solutions.

scholarly journals Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations

2019 ◽  
Vol 40 (4) ◽  
pp. 2473-2504 ◽  
Author(s):  
Claire Chainais-Hillairet ◽  
Maxime Herda
Keyword(s):  
Finite Volume ◽  
Exponential Decay ◽  
Large Time ◽  
Nonlinear Models ◽  
Numerical Test ◽  
Diffusion Equations ◽  
Time Behaviour ◽  
Finite Volume Schemes ◽  
Large Time Behaviour ◽  
Convection Diffusion

Abstract We are interested in the large-time behaviour of solutions to finite volume discretizations of convection–diffusion equations or systems endowed with nonhomogeneous Dirichlet- and Neumann-type boundary conditions. Our results concern various linear and nonlinear models such as Fokker–Planck equations, porous media equations or drift–diffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. In this paper we show that in the framework of finite volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserves this exponential decay of the discrete solution to the discrete steady state of the scheme. This includes for instance upwind and centred convections or Scharfetter–Gummel discretizations. We illustrate our theoretical results on several numerical test cases.

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