Numeric solution of non-linear hyperbolic PDE’s: Study of the shallow water equations

Abstract

In this project, the numeric solution of a system of non-linear hyperbolic PDE’s is analyzed. Particularly, the conservation laws for the shallow water equations for the dam-break problem are investigated. Godunov, McCormack and ENO methods, first, second and third order numerical schemes respectively, have been developed in order to solve problems with discontinuities (shocks), e.g., the dam-break and tsunami problems. In order to study the accuracy and convergence of the methods, three numerical examples have been studied. Firstly, the one-dimensional case of the dam-break problem. Secondly, the two-dimensional case but with the slip wall boundary conditions in the y-direction. With this approach, the numerical scheme can be validated with the 1D analytical solution and the e↵ect of di↵erent mesh sizes and time increments on the accuracy and computational cost of the method can be analyzed. Then, a 2D problem where the shock waves propagate in both directions is also investigated. This method will be combined with the immersed boundary method to investigate tsunami propagation in the coastal regions in the future.