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1.	Hadi Shamsnia, S., et al. (author) A comparative study of two different shallow water formulations using stable summation by parts schemes 2022 In: Wave motion. - : Elsevier BV. - 0165-2125 .- 1878-433X. ; 114 Journal article (peer-reviewed)abstract 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.

Hadi Shamsnia, S., et al. (author)
A comparative study of two different shallow water formulations using stable summation by parts schemes
2022
In: Wave motion. - : Elsevier BV. - 0165-2125 .- 1878-433X. ; 114
Journal article (peer-reviewed)abstract
- 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.

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