| Citation: |
Guenbo Hwang. ANALYTICAL SOLUTION FOR THE TWO-DIMENSIONAL LINEAR ADVECTION-DISPERSION EQUATION IN POROUS MEDIA VIA THE FOKAS METHOD[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2334-2354. doi: 10.11948/20200383
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ANALYTICAL SOLUTION FOR THE TWO-DIMENSIONAL LINEAR ADVECTION-DISPERSION EQUATION IN POROUS MEDIA VIA THE FOKAS METHOD
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Abstract
We present the analytical solution of the two-dimensional linear advection-dispersion equation ($ 2 $-D LAD) in the quarter plane and the semi-infinite domain for two-dimensional solute transport in a porous medium. The governing equation includes terms describing advection, longitudinal and transverse dispersions and linear equilibrium adsorption. The analytical solution in terms of integrals in the complex plane is established by utilizing the unified transform method, also known as the Fokas method. The method hinges upon analysis of the divergence form of the governing equation and the so-called global relation, which is an algebraic relation coupling all known and unknown initial and boundary values. Particularly, the integral representation of the solution yields an accurate and fast numerical evaluation of the solution for the $ 2 $-D LAD equation. We demonstrate examples as an application of the developed solution and compare the analytical solution with numerical results.
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References
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Guenbo Hwang. ANALYTICAL SOLUTION FOR THE TWO-DIMENSIONAL LINEAR ADVECTION-DISPERSION EQUATION IN POROUS MEDIA VIA THE FOKAS METHOD[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2334-2354. doi: 10.11948/20200383
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Figure 1. (Left) The region
$ D_1 = D_1^+\cup D_1^- $ (shaded) in the complex$ k_1 $ -plane, where$ {\mathop{{\rm{Re}}}\nolimits} \omega_1(k_1) <0 $ with$ \omega_1(k_1) = b k_1^2+ik_1 $ . (Right) The region$ D_2 = D_2^+\cup D_2^- $ (shaded) in the complex$ k_2 $ -plane, where$ {\mathop{{\rm{Re}}}\nolimits} \omega_2(k_2) <0 $ with$ \omega_2(k_2) = a k_2^2 $ . -
Figure 2. (Left) Efficient contour
$ L = L_1\cup L_2 $ for numerical integration. (Right) Efficient contour$ \Gamma = \Gamma_1\cup \Gamma_2\cup \Gamma_\epsilon $ for numerical integration (see the text for some details). -
Figure 4. Exponential convergence of the Fokas method for the case of the point pulse initial value. Relative error in the
$ L^2 $ norm versus the number of quadrature points$ N $ when (Left)$ t = 5 $ and (Right)$ t = 10 $ under different dispersion coefficients. -
Figure 3. (a, c, e) Comparison of the analytical solutions (solid curves) and the numerical solutions (symbols) with the point pulse initial value in a longitudinal cross-section at
$ y = 0 $ under the different dispersion coefficients. (b, d, f) Analytical solution profiles over the$ xt $ -plane, where$ y = 0 $ under the different dispersion coefficients. -
Figure 6. Exponential convergence of the Fokas method for the case of Gaussian initial value. Relative error in the
$ L^2 $ norm versus the number of quadrature points$ N $ when (Left)$ t = 5 $ and (Right)$ t = 10 $ under different dispersion coefficients. -
Figure 5. (a, c, e) Comparison of the analytical solutions (solid curves) and the numerical solutions (symbols) with the Gaussian initial value in a longitudinal cross-section at
$ y = 0 $ under the different dispersion coefficients. (b, d, f) Analytical solution profiles over the$ xt $ -plane, where$ y = 0 $ under the different dispersion coefficients.