A new well-balanced finite-volume scheme on unstructured triangular grids for two-dimensional two-layer shallow water flows with wet-dry fronts

AbstractA finite-volume numerical method for studying shallow water flows over an arbitrary bed profile in the presence of external force has been proposed in [33]. This method uses the quasi-two-layer model of hydrodynamic flows over a stepwise boundary with advanced consideration of the flow features near the step. A distinctive feature of the suggested model is a separation of the studied flow into two layers in calculating the flow quantities near each step, and improving by this means the approximation of depth-averaged solutions of the initial three-dimensional Euler equations. We are solving the shallow-water equations for one layer, introducing the fictitious lower layer only as an auxiliary structure in setting up the appropriate Riemann problems for the upper layer. Besides, the quasi-two-layer approach leads to the appearance of additional terms in the one-layer finite-difference representation of balance equations. Numerical simulations are performed based on the proposed in [33] algorithm of various physical phenomena, such as breakdown of the rectangular fluid column over an inclined plane, large-scale motion of fluid in the gravity field in the presence of Coriolis force over amounted obstacle on the underlying surface. Computations are made for the two-dimensional dam-break problem on a slope precisely conform to laboratory experiments. The interaction of the Tsunami wave with the shore line including an obstacle has been simulated to demonstrate the efficiency of the developed algorithm in domains, including partly flooded and dry regions.

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An entropy stable finite volume scheme for the two dimensional Navier–Stokes equations on triangular grids

Applied Mathematics and Computation ◽

10.1016/j.amc.2017.07.020 ◽

2017 ◽

Vol 314 ◽

pp. 257-286 ◽

Cited By ~ 1

Author(s):

Deep Ray ◽

Praveen Chandrashekar

Keyword(s):

Finite Volume ◽

Stokes Equations ◽

Navier Stokes ◽

Finite Volume Scheme ◽

Two Dimensional ◽

Navier Stokes Equations ◽

Volume Scheme ◽

Triangular Grids ◽

Entropy Stable

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On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

Journal of Scientific Computing ◽

10.1007/s10915-021-01521-z ◽

2021 ◽

Vol 88 (1) ◽

Author(s):

Saray Busto ◽

Michael Dumbser ◽

Sergey Gavrilyuk ◽

Kseniya Ivanova

Keyword(s):

Numerical Methods ◽

Shallow Water ◽

Discontinuous Galerkin ◽

Finite Volume ◽

Finite Volume Scheme ◽

Vanishing Viscosity ◽

Vanishing Viscosity Limit ◽

Water Flows ◽

Shallow Water Flows ◽

Viscosity Limit

AbstractIn this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

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