Numerical solution of the advection-diffusion equation using Laplace transform finite analytical method

In this study, a numerical method has been investigated and developed to solve the one-dimensional advection-diffusion equation to predict the quality of water in rivers. In this method, time variable is eliminated first by the Laplace transformation, and then a finite analytical method is applied in space. Both are based on a local element of the discretized domain in a finite-volume method. Since the Laplace transformation has been used for temporal approximation, an efficient and accurate inverse Laplace transform method of De Hoog 1982 [An improved method for numerical inversion of Laplace transform. SIAM, Journal of Scientific and Statistical Computing, 3 (3), 357-366] is employed to obtain the solution in real time. The proposed method is compared against analytical solutions and two finite-difference methods. The present computations and comparisons show that the proposed method is superior to the finite-difference methods. The results of the proposed method also agree with analytical solutions without numerical oscillation or diffusion. The present method is applied to steady and unsteady flows and it also provides flexibility for uniform and non-uniform grid spacing and for a wide range of Peclet numbers. It takes less computational effort than finite-difference methods.