Adaptive modelling of two-dimensional shallow water flows with wetting and drying (original) (raw)
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Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management
This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated-unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet-Neumann method, using the dune-grid-glue module in the numerics software DUNE.
International Journal for Numerical Methods in Fluids, 2008
We build and analyze a Runge-Kutta Discontinuous Galerkin method to approximate the one-and two-dimensional Shallow-Water Equations. We introduce a flux modification technique to derive a wellbalanced scheme preserving steady-states at rest with variable bathymetry and a slope modification technique to deal satisfactorily with flooding and drying. Numerical results illustrating the performance of the proposed scheme are presented.
Computer Methods in Applied Mechanics and Engineering, 2009
This paper proposes a wetting and drying treatment for the piecewise linear Runge-Kutta discontinuous Galerkin approximation to the shallow water equations. The method takes a fixed mesh approach as opposed to mesh adaptation techniques and applies a post-processing operator to ensure the positivity of the mean water depth within each finite element. In addition, special treatments are applied in the numerical flux computation to prevent an instability due to excessive drying. The proposed wetting and drying treatment is verified through comparisons with exact solutions and convergence rates are examined. The obtained orders of convergence are close to or approximately equal to 1 for solutions with discontinuities and are improved for smooth solutions. The combination of the proposed wetting and drying treatment and a TVB slope limiter is also tested and is found to be applicable on condition that they are applied exclusively to an element at the same Runge-Kutta step.
KSCE Journal of Civil Engineering, 2015
The wet-dry scheme for moving boundary treatment is implemented in the framework of discontinuous Galerkin shallow water equations. As a formulation of approximate Riemann solver, the HLL (Harten-Lax-van Leer) numerical fluxes are employed and the TVB (Total Variation Bounded) slope limiter is used for the removal of unnecessary oscillations. As benchmark test problems, the dam-break problems and the classical problem of periodic oscillation in the parabolic bowl are solved with linear triangular elements and second-order Runge-Kutta scheme. The results are compared with exact solutions and the numerical solutions of previous study. For a more practical application, the implicit Runge-Kutta scheme is employed for the bottom friction terms and the moving shoreline in a rectangular basin of varying slopes is simulated. In all case studies, good agreement is observed with exact solutions or other well-known numerical solutions.
An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods
International journal for …, 2006
In this paper, we present a Discontinuous Galerkin formulation of the shallow water equations. An orthogonal basis is used for the spatial discretization and an explicit Runge-Kutta scheme is used for time discretization. Some results of second order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps.
Adaptive discontinuous Galerkin method for the shallow water equations
International Journal for …, 2006
In this paper, we present a Discontinuous Galerkin formulation of the shallow water equations. An orthogonal basis is used for the spatial discretization and an explicit Runge-Kutta scheme is used for time discretization. Some results of second order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps.
The Lagrange-Galerkin method for the two-dimensional shallow water equations on adaptive grids
International Journal for Numerical Methods in Fluids, 2000
The weak Lagrange-Galerkin finite element method for the two-dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non-linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self-adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange-Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second-order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant-Friedrich-Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange-Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid-point rule and a general family of Runge-Kutta schemes. Results for the two-dimensional advection equation with and without time-dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two-dimensional shallow water equations on a non-linear soliton wave are presented to illustrate the power and flexibility of this strategy.
Flooding and drying in discontinuous Galerkin discretizations of shallow water equations
Automatica, 2006
Accurate modeling of flooding and drying is important in forecasting river floods and near-shore hydrodynamics. We consider the space-time discontinuous Galerkin finite element discretization for shallow water equations with linear approximations of flow field. In which, the means (zeroth order approximation) is used to conserve the mass and momentum, and the slopes (first order approximation) are used to capture the front movement accurately in contrast to the finite volume schemes, where the slopes have to be reconstructed. As a preliminary step, we specify the front movement from some available exact solutions and show that the numerical results are second order accurate for linear polynomials. To resolve the front movement accurately in the context of discontinuous Galerkin discretizations, the front tracking and the front capturing methods are currently under investigation.
Computer Methods in Applied Mechanics and Engineering, 2013
To simulate complex flows involving wet-dry fronts in irregular terrains over arbitrary beds, this paper presents a 2D well-balanced shallow water flow model, based on an unstructured cell-centered finite volume scheme. In this model, hydrostatic reconstruction is applied to reconstruct non-negative water depths at wet-dry interfaces. Harten, Lax and van Leer approximate Riemann solver with the contact wave restored is employed to compute the fluxes of mass and momentum. In addition, the splitting point-implicit method is utilized to solve the friction source terms. The novel aspects of the model include the adaptive method and the slope source term treatment. The former is devised to prevent unphysical high velocities occurring in the part of the domain with varying thin water and topography. The latter converts the slope source terms of a cell into fluxes through its edges and takes account of the influence of wet-dry fronts, so as to satisfy the C-property in any case on unstructured grids. The accuracy and robustness of the proposed model are extensively verified against several benchmark tests as well as a real dam-break event, where its superiority in simulating complex flows with wetting and drying over uneven bed is emphasized.