Short-Term and Long-Term Solutions of Two-Dimensional Shallow Water Equations Using the Modified Decomposition Method

Abstract

In this paper, two modifications of the decomposition method (DM) including its multistage form (say here MSDM) and its combination with the Pade approximants (say here DMPA) are proposed and applied for the short-term and long-term solutions of two-dimensional shallow water equations (SWEs) including friction effects. Due to nonlinear behavior of the model, the DM is used to facilitate the computations on the nonlinear terms. However, the MS technique is used to obtain a solution for longer periods of time and to prevent accuracy issues associated with the standard DM at large times. The results of the proposed method are compared with those obtained using the PAs as an after treatment technique which also yields an accurate solution in our work. It has been shown that the MS technique can efficiently be used in the solution of the nonlinear coupled SWEs and would extend the domain of validity of the solution obtained via the standard DM. However, the necessity of predicting the long-term behavior of flow in some applications makes the PAs superior to the MS technique with respect to accuracy and CPU time. Numerical and graphical results of the flow characteristics for a given application of two-dimensional flood wave over two types of mild and steep slopes are presented to elucidate the efficiency of the proposed modifications in the study of the flow behavior. Moreover, a convergence analysis and the time evolution of the residuals as an index of the solution error are also presented in tabular and graphical forms.