Computer Methods in Applied Mechanics and Engineering - Special edition on the 20th Anniversary
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
SIAM Journal on Numerical Analysis
Spectral methods on triangles and other domains
Journal of Scientific Computing
Genuinely multidimensional upwinding for the 2D shallow water equations
Journal of Computational Physics
A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
SIAM Journal on Numerical Analysis
Conservative multidimensional upwinding for the steady two-dimensional shallow water equations
Journal of Computational Physics
Journal of Computational Physics
Flux difference splitting and the balancing of source terms and flux gradients
Journal of Computational Physics
Toward the ultimate conservative scheme: following the quest
Journal of Computational Physics
A smoothness indicator for adaptive algorithms for hyperbolic systems
Journal of Computational Physics
Residual Distribution Schemes for Conservation Laws via Adaptive Quadrature
SIAM Journal on Scientific Computing
Journal of Computational Physics
High Order Fluctuation Schemes on Triangular Meshes
Journal of Scientific Computing
Journal of Computational Physics
A second-order-accurate monotone implicit fluctuation splitting scheme for unsteady problems
Journal of Computational Physics
Journal of Computational Physics
Residual distribution for general time-dependent conservation laws
Journal of Computational Physics
Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems
Journal of Computational Physics
Third-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems
Journal of Computational Physics
A new Savage-Hutter type model for submarine avalanches and generated tsunami
Journal of Computational Physics
Discontinuous fluctuation distribution
Journal of Computational Physics
Stabilized residual distribution for shallow water simulations
Journal of Computational Physics
High-order well-balanced schemes and applications to non-equilibrium flow
Journal of Computational Physics
A first-order system approach for diffusion equation. II: Unification of advection and diffusion
Journal of Computational Physics
Explicit Runge-Kutta residual distribution schemes for time dependent problems: Second order case
Journal of Computational Physics
Journal of Scientific Computing
Journal of Scientific Computing
A Hybrid Second Order Scheme for Shallow Water Flows
Journal of Scientific Computing
Hybrid Second Order Schemes for Scalar Balance Laws
Journal of Scientific Computing
A numerical treatment of wet/dry zones in well-balanced hybrid schemes for shallow water flow
Applied Numerical Mathematics
Well-Balanced Adaptive Mesh Refinement for shallow water flows
Journal of Computational Physics
Hi-index | 31.49 |
We consider the numerical solution of the shallow water equations on unstructured grids. We focus on flows over wet areas. The extension to the case of dry bed will be reported elsewhere. The shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding, in this case, with the total energy. Our aim is to show the application of a particular class of conservative residual distribution (RD) schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties. We give a review of conservative RD schemes showing relations between different approaches previously published, and recall L^~ stability and accuracy criteria characterizing the schemes. In particular, the accuracy of the RD method in presence of source terms is analyzed, and conditions to construct rth order discretizations on irregular triangular grids are proved. It is shown that the RD approach gives a natural way of obtaining high order discretizations which, moreover, preserves exactly the steady lake at rest solution independently on mesh topology, nature of the variation of the bottom and polynomial order of interpolation used for the unknowns. We also consider more general analytical solutions which are less investigated from the numerical view point. On irregular grids, linearity preserving RD schemes yield a truly second order approximation. We also sketch a strategy to achieve discretizations which preserve exactly some of these solutions. Numerical results on steady and time-dependent problems involving smooth and non-smooth variations of the bottom topology show very promising features of the approach.