A comparative study of two different shallow water formulations using stable summation by parts schemes

• Published in: Wave motion 2022-09, Vol.114, p.103043, Article 103043
• Main Authors: Hadi Shamsnia, S., Ghader, Sarmad, Abbas Haghshenas, S., Nordström, Jan
• Format: Article
• Language: English
• Subjects: Divergence; Finite difference method; High order accuracy; Open boundary; Penalty terms; Shallow water equations; Summation by parts; Vorticity; Weak boundary conditions
• ISSN: 0165-2125; 1878-433X; 1878-433X
• DOI: 10.1016/j.wavemoti.2022.103043

This study provides numerical solutions to the two-dimensional linearized shallow water equations (SWE) using a high-order finite difference scheme in Summation By Parts (SBP) form. In addition to the SBP operators for the discretizations, penalty terms, Simultaneous Approximation Terms (SAT) are applied to impose well-posed open boundary conditions. The conventional SWE with height and velocities as the prognostic variables, and a new type of the vorticity–divergence SWE with wave height gradients, vorticity and divergence as the prognostic variables were investigated. It was shown that the solution in all numerical tests enter and exit the domain without instabilities. The convergence rates were correct for all orders of the SBP operators in both the entrance and exit tests. Interestingly, the error norm of the wave height were orders of magnitude lower in the vorticity–divergence solutions compared to the conventional SWE solutions. •A numerical solution to two-dimensional shallow water equations is presented.•Two different formulation of the shallow water equations is considered.•High-order finite difference scheme on summation by parts is applied.•Simultaneous approximation terms are used to impose well-posed stable BC.•The numerical results are shown to be convergent and reasonable.