ADER-WAF Schemes for the Homogeneous One-Dimensional Shallow Water Equations

ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is $C_{\text{CFL}}\le 1$ , where $C_{\text{CFL}}$ denotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed $C_{\text{CFL}}=0.95$ for their experiments. Nonetheless, we noted that the experiments conducted in this study with $C_{\text{CFL}}=0.95$ produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.