scholarly journals Numerical solution of advection-diffusion equation using finite difference schemes

2020 ◽  
Vol 55 (1) ◽  
pp. 15-22
Author(s):  
LS Andallah ◽  
MR Khatun
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Finite Difference Schemes ◽  
Qualitative Behavior ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nicolson Scheme ◽  
And Diffusion ◽  
Crank Nicolson

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

scholarly journals Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
A. R. Appadu
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Finite Difference Schemes ◽  
Time Step ◽  
Partial Differential ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Dispersion And Dissipation Errors ◽  
Nicolson Scheme ◽  
Nonstandard Finite Difference

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted byhandk, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advection-diffusion equation at some values ofkandh. Two optimisation techniques are then implemented to find the optimal values ofkwhenh=0.02for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.

scholarly journals Time-Splitting Procedures for the Numerical Solution of the 2D Advection-Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-20 ◽  
Author(s):  
A. R. Appadu ◽  
H. H. Gidey
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Optimization Technique ◽  
Step Size ◽  
One Dimensional ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Optimal Value ◽  
Time Splitting ◽  
Dispersion And Dissipation Errors

We perform a spectral analysis of the dispersive and dissipative properties of two time-splitting procedures, namely, locally one-dimensional (LOD) Lax-Wendroff and LOD (1, 5) [9] for the numerical solution of the 2D advection-diffusion equation. We solve a 2D numerical experiment described by an advection-diffusion partial differential equation with specified initial and boundary conditions for which the exact solution is known. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion and dissipation errors. Lastly, an optimization technique is implemented to find the optimal value of temporal step size that minimizes the dispersion error for both schemes when the spatial step is chosen as 0.025, and this is validated by numerical experiments.

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