J. Murillo

Weak solutions for partial differential equations with source terms: Application to the shallow water equations

Weak solutions of problems with m equations with source terms are proposed using an augmented Riemann solver defined by m+1 states instead of increasing the number of involved equations. These weak solutions use propagating jump discontinuities connecting the m+1 states to approximate the Riemann solution. The average of the propagated waves in the computational cell leads to a reinterpretation of the Roes approach and in the upwind treatment of the source term of Vazquez-Cendon. It is derived that the numerical scheme can not be formulated evaluating the physical flux function at the position of the initial discontinuities, as usually done in the homogeneous case. Positivity requirements over the values of the intermediate states are the only way to control the global stability of the method. Also it is shown that the definition of well-balanced equilibrium in trivial cases is not sufficient to provide correct results: it is necessary to provide discrete evaluations of the source term that ensure energy dissipating solutions when demanded. The one and two dimensional shallow water equations with source terms due to the bottom topography and friction are presented as case study. The stability region is shown to differ from the one defined for the case without source terms, and it can be derived that the appearance of negative values of the thickness of the water layer in the proximity of the wet/dry front is a particular case, of the wet/wet fronts. The consequence is a severe reduction in the magnitude of the allowable time step size if compared with the one obtained for the homogeneous case. Starting from this result, 1D and 2D numerical schemes are developed for both quadrilateral and triangular grids, enforcing conservation and positivity over the solution, allowing computationally efficient simulations by means of a reconstruction technique for the inner states of the weak solution that allows a recovery of the time step size.

An Exner-based coupled model for two-dimensional transient flow over erodible bed

Transient flow over erodible bed is solved in this work assuming that the dynamics of the bed load problem is described by two mathematical models: the hydrodynamic model, assumed to be well formulated by means of the depth averaged shallow water equations, and the Exner equation. The Exner equation is written assuming that bed load transport is governed by a power law of the flow velocity and by a flow/sediment interaction parameter variable in time and space. The complete system is formed by four coupled partial differential equations and a genuinely Roe-type first order scheme has been used to solve it on triangular unstructured meshes. Exact solutions have been derived for the particular case of initial value Riemann problems with variable bed level and depending on particular forms of the solid discharge formula. The model, supplied with the corresponding solid transport formulae, is tested by comparing with the exact solutions. The model is validated against laboratory experimental data of different unsteady problems over erodible bed.

Augmented versions of the HLL and HLLC Riemann solvers including source terms in one and two dimensions for shallow flow applications

Shallow water flows are found in a variety of engineering problems always dominated by the presence of bed friction and irregular bathymetry. These source terms determine completely the possible evolution of the flooded area in time. It is well known that appropriate numerical schemes for this type of flows must be well-balanced. Well-balanced numerical schemes are based on the preservation of cases of quiescent equilibrium over variable bed elevation. Commonly they are formulated as an adaptation of numerical solvers defined for cases without source terms. This procedure is insufficient when applied to real situations. Then, it is possible to argue that appropriate numerical schemes cannot arise directly from those derived from the simplest homogeneous case without source terms. New solutions are presented in this work by defining weak solutions that include the presence of source terms. To do that, the solvers presented in this work extend the number of waves in the well known HLL and HLLC solvers involving a stationary jump in the solution. This is done without modifying the original solution vector of conserved quantities. The resulting approximate Riemann solvers include variable bed level surface and friction. Solvers are systematically assessed via a series of test problems with exact solutions for one and two dimensions, including steady and unsteady flow configurations, variation of the flooded area in time and comparisons with experimental data. The obtained results point out that the new method is able to predict faithfully the overall behavior of the solution and of any type of waves.

Conservative numerical simulation of multi-component transport in two-dimensional unsteady shallow water flow

An explicit finite volume model to simulate two-dimensional shallow water flow with multi-component transport is presented. The governing system of coupled conservation laws demands numerical techniques to avoid unrealistic values of the transported scalars that cannot be avoided by decreasing the size of the time step. The presence of non conservative products such as bed slope and friction terms, and other source terms like diffusion and reaction, can make necessary the reduction of the time step given by the Courant number. A suitable flux difference redistribution that prevents instability and ensures conservation at all times is used to deal with the non-conservative terms and becomes necessary in cases of transient boundaries over dry bed. The resulting method belongs to the category of well-balanced Roe schemes and is able to handle steady cases with flow in motion. Test cases with exact solution, including transient boundaries, bed slope, friction, and reaction terms are used to validate the numerical scheme. Laboratory experiments are used to validate the techniques when dealing with complex systems as the @k-@e model. The results of the proposed numerical schemes are compared with the ones obtained when using uncoupled formulations.

Wave Riemann description of friction terms in unsteady shallow flows: Application to water and mud/debris floods

In this work, the source term discretization in hyperbolic conservation laws with source terms is considered using an approximate augmented Riemann solver. The technique is applied to the shallow water equations with bed slope and friction terms with the focus on the friction discretization. The augmented Roe approximate Riemann solver provides a family of weak solutions for the shallow water equations, that are the basis of the upwind treatment of the source term. This has proved successful to explain and to avoid the appearance of instabilities and negative values of the thickness of the water layer in cases of variable bottom topography. Here, this strategy is extended to capture the peculiarities that may arise when defining more ambitious scenarios, that may include relevant stresses in cases of mud/debris flow. The conclusions of this analysis lead to the definition of an accurate and robust first order finite volume scheme, able to handle correctly transient problems considering frictional stresses in both clean water and debris flow, including in this last case a correct modelling of stopping conditions.

An optimized GPU implementation of a 2D free surface simulation model on unstructured meshes

A GPU implementation of a FV method for the 2D Shallow Water Equations is presented.Structured and unstructured meshes allow different implementations.NVIDIA C2070 GPU is compared against Intel Core 2 Quad Processor.The basic GPU implementation obtains between 20× and 30× of speed-up.Some strategies on the mesh order allow to double the performance, reaching 50×. This work is related with the implementation of a finite volume method to solve the 2D Shallow Water Equations on Graphic Processing Units (GPU). The strategy is fully oriented to work efficiently with unstructured meshes which are widely used in many fields of Engineering. Due to the design of the GPU cards, structured meshes are better suited to work with than unstructured meshes. In order to overcome this situation, some strategies are proposed and analyzed in terms of computational gain, by means of introducing certain ordering on the unstructured meshes. The necessity of performing the simulations using unstructured instead of structured meshes is also justified by means of some test cases with analytical solution.

Energy balance numerical schemes for shallow water equations with discontinuous topography

The well-balanced property that ensures quiescent equilibrium when solving the shallow-water equations with varying topography is extended in this work to ensure numerically a constant level of energy in steady cases with velocity when necessary. This is done in the context of augmented solvers that consider in their definition the presence of a discontinuous bed. In order to guarantee a constant energy state a proper integral approach of the bed source term is presented. This approach is systematically assessed via a series of steady test cases and Riemann problems including the resonance regime.

Accurate numerical modeling of 1D flow in channels with arbitrary shape. Application of the energy balanced property

This work focuses on the numerical treatment of 1D flow in channels with arbitrary shape using energy balanced arguments. The system of equations is defined using the mass and momentum conservation equations, allowing the resolution of hydraulic jumps where energy conservation arguments are not valid. When necessary, conservation of mechanical energy takes part actively in the numerical scheme when evaluating the source terms. The numerical scheme is based on an augmented Roe solver that involves the presence of source terms by adding an extra stationary wave. The characteristics of the numerical scheme include the energy balanced property, and being only first order accurate in time and space, leads to exact numerical solutions for steady solutions with independence of the grid refinement in channels with general geometries. Riemann problems considered here involve non-prismatic channels, bed variations and the resonance regime, including the limiting situation when the Riemann data belong to the resonance hypersurface. Numerical results point out that the finite volume numerical scheme with nonconservative terms presented here, converges to the exact solution. The well balanced property is ensured, as it is a particular case of the energy balanced property in cases of quiescent equilibrium.

Fertigation in Furrows and Level Furrow Systems. I: Model Description and Numerical Tests

The simulation of fertigation in furrows and level furrow systems faces a number of problems resulting in relevant restrictions to its widespread application. In this paper, a simulation model is proposed that addresses some of these problems by: (1) implementing an infiltration model that adjusts to the variations in wetted perimeter; (2) using a friction model that adjusts to different flows and which uses an absolute roughness parameter; (3) adopting an equation for the estimation of the longitudinal diffusion coefficient; and (4) implementing a second-order TVD numerical scheme and specific treatments for the boundary conditions and the junctions. The properties of the proposed model were demonstrated using three numerical tests focusing on the numerical scheme and the treatments. The application of the model to the simulation of furrows and furrow systems is presented in a companion paper, in which the usefulness of the innovative aspects of the proposed model is demonstrated.

Two-dimensional depth-averaged modelling of dam-break flows over mobile beds

The paper is aimed at the description and validation of a novel two-dimensional depth-averaged simulation tool for highly unsteady discontinuous flows over complex time-evolving geometries. The conceptual model is developed within the shallow-flow framework and features non-equilibrium sediment transport. A fully conservative finite-volume discretization scheme is employed. The treatment of bed-slope source terms ensures that the scheme is well balanced and leads to correct energy losses in discontinuities. Existing laboratory data are used to validate the model and to discuss some of its embedded formulations. Blind-test comparison shows that the model is capable of describing the main dam-break flow and bed morphology features. The solution is shown to be sensitive to the formulations for sediment transport capacity and for adaptation length (Λ). A better functional relation of Λ with the Shields parameter is investigated. The quality of the solution is shown to be only marginally improved by the inclusion of turbulent stresses.