| 1. |
- Ghader, Sarmad, et al.
(författare)
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High-order compact finite difference schemes for the spherical shallow water equations
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- This work is devoted to the application of the super compact finite difference (SCFDM) and the combined compact finite difference (CCFDM) methods for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence and height. Five high-order schemes including the fourth-order compact, the sixth-order and eighth-order SCFDM and the sixth-order and eighth-order CCFDM schemes are used for spatial differencing of the spherical shallow water equations. To advance the solution in time, a semi-implicit Runge-Kutta method is used. In addition, to control the nonlinear instability and avoiding the polar problem a high-order spatial filter is proposed. An unstable barotropic mid-latitude zonal jet is employed as an initial condition. For the numerical solution of the elliptic equations in the problem, a direct hybrid method which consists of using a high-order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate is utilized. The convergence rate for all methods is studied and veried. Qualitative and quantitative assessment of the results, such as measures of maximum vorticity gradient, power spectrum of total energy, relative change in potential enstrophy and potential palinstrophy, reveal that the sixth-order and eighth-order CCFDM and the sixth-order and eighth-order SCFDM methods lead to a remarkable improvement of the solution over the fourth-order compact method. It is also shown that the performance of the sixth-order and eighth-order CCFDM methods are superior to the sixth-order and eighth-order SCFDM methods. Copyright c 2013 John Wiley & Sons, Ltd.
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| 2. |
- Ghader, Sarmad, et al.
(författare)
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High-order Compact Finite Difference Schemes for the Vorticity-divergence Representation of the Spherical Shallow Water Equations
- 2015
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Ingår i: International Journal for Numerical Methods in Fluids. - : John Wiley & Sons. - 0271-2091 .- 1097-0363. ; 78:12, s. 709-738
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Tidskriftsartikel (refereegranskat)abstract
- This work is devoted to the application of the super compact finite difference method (SCFDM) and the combined compact finite difference method (CCFDM) for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence, and height. The fourth-order compact, the sixth-order and eighth-order SCFDM, and the sixth-order and eighth-order CCFDM schemes are used for the spatial differencing. To advance the solution in time, a semi-implicit Runge–Kutta method is used. In addition, to control the nonlinear instability, an eighth-order compact spatial filter is employed. For the numerical solution of the elliptic equations in the problem, a direct hybrid method, which consists of a high-order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate, is utilized. The accuracy and convergence rate for all methods are verified against exact analytical solutions. Qualitative and quantitative assessments of the results for an unstable barotropic mid-latitude zonal jet employed as an initial condition are addressed. It is revealed that the sixth-order and eighth-order CCFDMs and SCFDMs lead to a remarkable improvement of the solution over the fourth-order compact method.
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| 3. |
- Ghader, Sarmad, et al.
(författare)
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Revisiting well-posed boundary conditions for the shallow water equations
- 2014
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- A general form of well-posed open boundary conditions for the two-dimensional shallow water equations is derived. To this end, the energy method, in which one bounds the energy of the solution by choosing a minimal number of suitable boundary conditions, is used. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. The results are compared with a number of often used open boundary conditions and it is shown that they are a subset of the derived general form.
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| 4. |
- Ghader, Sarmad, et al.
(författare)
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Revisiting well-posed boundary conditions for the shallow water equations
- 2014
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Ingår i: Dynamics of atmospheres and oceans (Print). - : Elsevier. - 0377-0265 .- 1872-6879. ; 66, s. 1-9
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Tidskriftsartikel (refereegranskat)abstract
- We derive a general form of well-posed open boundary conditions for the two-dimensional shallow water equations by using the energy method. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. The boundary conditions are also discussed for a rectangular domain. We compare the results with a number of often used open boundary conditions and show that they are a subset of the derived general form.
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| 5. |
- Ghader, Sarmad, et al.
(författare)
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Well-posed boundary conditions for the shallow water equations
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We derive well-posed boundary conditions for the two-dimensional shallow water equations by using the energy method. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. Then, as an example, the boundary conditions are discussed for a rectangular domain.
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| 6. |
- Hadi Shamsnia, S., et al.
(författare)
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A comparative study of two different shallow water formulations using stable summation by parts schemes
- 2022
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Ingår i: Wave motion. - : Elsevier BV. - 0165-2125 .- 1878-433X. ; 114
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Tidskriftsartikel (refereegranskat)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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| 7. |
- Nordström, Jan, et al.
(författare)
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A New Well-posed Vorticity Divergence Formulation of the Shallow Water Equations
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A completely new vorticity-divergence formulation of the two-dimensional shallow water equations including boundary conditions is derived. The new formulation is necessary since the conventional one does not lead to a well-posed initial boundary value problem for limited area modelling. The new vorticity-divergence formulation include four dependent variables instead of three, and require more equations and boundary conditions than the conventional formulation. On the other hand, it forms a symmetrizable hyperbolic set of equations with well defined boundary conditions that leads to a well-posed problem with a bounded energy.
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| 8. |
- Nordström, Jan, et al.
(författare)
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A New Well-posed Vorticity Divergence Formulation of the Shallow Water Equations
- 2015
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Ingår i: Ocean Modelling. - : Elsevier. - 1463-5003 .- 1463-5011. ; 93, s. 1-6
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Tidskriftsartikel (refereegranskat)abstract
- A new vorticity–divergence formulation of the two-dimensional shallow water equations including boundary conditions is derived. The new formulation is necessary since the conventional one does not lead to a well-posed initial boundary value problem for limited-area modelling.The new vorticity–divergence formulation includes four dependent variables instead of three and requires more equations and boundary conditions than the conventional formulation. On the other hand, it forms a hyperbolic set of equations with well-defined boundary conditions that leads to a well-posed problem with bounded energy.
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