Impact of the Finite Element Mesh Structure on the Solution Accuracy of a Two-Dimensional Kinematic Wave Equation

This paper presents the influence of the finite element mesh structure on the accuracy of the numerical solution of a two-dimensional linear kinematic wave equation. This equation was solved using a two-level scheme for time integration and a modified finite element method with triangular elements for space discretization. The accuracy analysis of the applied scheme was performed using a modified equation method for three different uniform triangular meshes with the same resolution, but with a different structure. The modified equation approach based on the Taylor series truncation allowed the numerical diffusivity and dispersivity tensors to be derived, which are directly associated with numerical errors. The derived tensors depend on parameters such as the space and time interval, flow velocity, and weighting coefficients. A detailed analysis carried out for the particular values of these parameters enabled an assessment of the numerical errors that may be generated in the solution for the assumed mesh structure. The theoretical analysis was confirmed by using numerical simulations carried out for an arbitrary domain and auxiliary conditions. According to the obtained results, it appears that it is possible to improve the accuracy of the numerical solution by choosing the proper mesh structure and numerical parameters for the applied algorithm.