Time discretization for modeling migration of groundwater contaminant in the presence of micro-organisms via a semi-analytic method

The parabolic systems for migration of groundwater contaminants are studied through the semi-analytic method (SAM). For systems, a modification of the classic trigonometric basis function (TBF) is proposed that successfully obtains the solution on the available boundary data and improves accuracy. The method is based on a theory that removes instability and keeps the same accuracy. However, especially when using SAM schemes with acceptable stability properties, one is still faced with the considerable task of linearizing the systems, consequently, the linearization of the systems which approximate the nonlinear term with the first order derivative is introduced. The final approximation is given by the summation of the primary approximation, radial basis functions (RBFs), and the related TBFs which are determined by the homogeneous boundary conditions. Then the approximation is substituted back to the governing equations where the unknown coefficients can be determined. The efficiency of the algorithm is highlighted by numerical simulations relating to a model on test cases with analytical solutions and without an analytical solution.