An entropy stable discontinuous Galerkin method for the two-layer shallow water equations on curvilinear meshes. (English) Zbl 1533.65191
The document addresses the development of a numerical method to solve the two-layer shallow water equations, which are fundamental in modelling the dynamics of two immiscible fluid layers of varying densities under shallow water assumption. The authors propose an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) formulated on curvilinear meshes, leveraging the properties of Legendre-Gauss-Lobatto nodes for high-order, path-conservative, and entropy conservative approximations.
The scientific problem tackled in the paper revolves around the computational challenges associated with the two-layer shallow water system, particularly the difficulties arising from nonconservative products that affect wave propagation, making the system only conditionally hyperbolic. The authors’ method aims to ensure entropy stability, a well-balanced nature for discontinuous bathymetry, and the preservation of steady-state solutions, which are critical for accurate and stable numerical simulations of such fluid dynamics problems.
Methodologically, the paper details the construction of the DGSEM, emphasizing the use of summation-by-parts property, flux differencing formulation, and specific combinations of numerical fluxes and discretization strategies for nonconservative terms to achieve entropy conservation and stability. This approach ensures the method’s applicability to complex domain geometries and its robustness in handling discontinuities and maintaining the well-balanced property.
The main findings of the study include the verification of theoretical properties through numerical tests, demonstrating convergence, entropy stability, and the well-balanced nature of the scheme. The research contributes significantly to the field of computational fluid dynamics by providing a robust and accurate method for simulating two-layer shallow water flows over curvilinear domains, with potential applications ranging from environmental to engineering problems involving fluid interfaces.
This paper’s significance lies in its advancement of numerical methods for fluid dynamics, offering a tool that combines high-order accuracy with the critical capability to respect the physical laws of entropy and mass conservation in challenging scenarios of fluid dynamics.
The scientific problem tackled in the paper revolves around the computational challenges associated with the two-layer shallow water system, particularly the difficulties arising from nonconservative products that affect wave propagation, making the system only conditionally hyperbolic. The authors’ method aims to ensure entropy stability, a well-balanced nature for discontinuous bathymetry, and the preservation of steady-state solutions, which are critical for accurate and stable numerical simulations of such fluid dynamics problems.
Methodologically, the paper details the construction of the DGSEM, emphasizing the use of summation-by-parts property, flux differencing formulation, and specific combinations of numerical fluxes and discretization strategies for nonconservative terms to achieve entropy conservation and stability. This approach ensures the method’s applicability to complex domain geometries and its robustness in handling discontinuities and maintaining the well-balanced property.
The main findings of the study include the verification of theoretical properties through numerical tests, demonstrating convergence, entropy stability, and the well-balanced nature of the scheme. The research contributes significantly to the field of computational fluid dynamics by providing a robust and accurate method for simulating two-layer shallow water flows over curvilinear domains, with potential applications ranging from environmental to engineering problems involving fluid interfaces.
This paper’s significance lies in its advancement of numerical methods for fluid dynamics, offering a tool that combines high-order accuracy with the critical capability to respect the physical laws of entropy and mass conservation in challenging scenarios of fluid dynamics.
Reviewer: Denys Dutykh (Le Bourget-du-Lac)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65D32 | Numerical quadrature and cubature formulas |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76T06 | Liquid-liquid two component flows |
| 86A05 | Hydrology, hydrography, oceanography |
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 35L50 | Initial-boundary value problems for first-order hyperbolic systems |
| 35Q30 | Navier-Stokes equations |
Keywords:
two-layer shallow water system; well-balanced method; discontinuous Galerkin spectral element method; entropy stabilitySoftware:
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