TITLE:
A Numerical Study of Riemann Problem Solutions for the Homogeneous One-Dimensional Shallow Water Equations
AUTHORS:
Pavlos Stampolidis, Maria Ch. Gousidou-Koutita
KEYWORDS:
1D Shallow Water Equations, Finite Volume WENO Schemes, Multi-Resolution WENO Schemes, Random Choice Method, Riemann Problem
JOURNAL NAME:
Applied Mathematics,
Vol.15 No.11,
November
21,
2024
ABSTRACT: The solution of the Riemann Problem (RP) for the one-dimensional (1D) non-linear Shallow Water Equations (SWEs) is known to produce four potential wave patterns for the scenario where the water depth is always positive. In this paper, we choose four test problems with exact solutions for the 1D SWEs. Each test problem is a RP with one of the four possible wave patterns as its solution. These problems are numerically solved using schemes from the family of Weighted Essentially Non-Oscillatory (WENO) methods. For comparison purposes, we also include results obtained from the Random Choice Method (RCM). This study has three main objectives. Firstly, we outline the procedures for the implementation of the methods employed in this paper. Secondly, we assess the performance of the schemes in conjunction with a second-order Total Variation Diminishing (TVD) flux on a variety of RPs for the 1D SWEs (for both short- and long-time simulations). Thirdly, we investigate if a single method yields optimal outcomes for all test problems. Optimal outcomes refer to numerical solutions devoid of spurious oscillations, exhibiting high resolution of discontinuities, and attaining high-order accuracy in the smooth parts of the solution.