1,230 research outputs found
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
In this paper, we introduce a discontinuous Finite Element formulation on
simplicial unstructured meshes for the study of free surface flows based on the
fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a
new class of asymptotically equivalent equations, which have a simplified
analytical structure, we consider a decoupling strategy: we approximate the
solutions of the classical shallow water equations supplemented with a source
term globally accounting for the non-hydrostatic effects and we show that this
source term can be computed through the resolution of scalar elliptic
second-order sub-problems. The assets of the proposed discrete formulation are:
(i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary
order of approximation in space, (iii) the exact preservation of the motionless
steady states, (iv) the preservation of the water height positivity, (v) a
simple way to enhance any numerical code based on the nonlinear shallow water
equations. The resulting numerical model is validated through several
benchmarks involving nonlinear wave transformations and run-up over complex
topographies
The VOLNA code for the numerical modelling of tsunami waves: generation, propagation and inundation
A novel tool for tsunami wave modelling is presented. This tool has the
potential of being used for operational purposes: indeed, the numerical code
\VOLNA is able to handle the complete life-cycle of a tsunami (generation,
propagation and run-up along the coast). The algorithm works on unstructured
triangular meshes and thus can be run in arbitrary complex domains. This paper
contains the detailed description of the finite volume scheme implemented in
the code. The numerical treatment of the wet/dry transition is explained. This
point is crucial for accurate run-up/run-down computations. Most existing
tsunami codes use semi-empirical techniques at this stage, which are not always
sufficient for tsunami hazard mitigation. Indeed the decision to evacuate
inhabitants is based on inundation maps which are produced with this type of
numerical tools. We present several realistic test cases that partially
validate our algorithm. Comparisons with analytical solutions and experimental
data are performed. Finally the main conclusions are outlined and the
perspectives for future research presented.Comment: 47 pages, 27 figures. Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh
A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids
A novel wetting and drying treatment for second-order Runge-Kutta
discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water
equations is proposed. It is developed for general conforming two-dimensional
triangular meshes and utilizes a slope limiting strategy to accurately model
inundation. The method features a non-destructive limiter, which concurrently
meets the requirements for linear stability and wetting and drying. It further
combines existing approaches for positivity preservation and well-balancing
with an innovative velocity-based limiting of the momentum. This limiting
controls spurious velocities in the vicinity of the wet/dry interface. It leads
to a computationally stable and robust scheme -- even on unstructured grids --
and allows for large time steps in combination with explicit time integrators.
The scheme comprises only one free parameter, to which it is not sensitive in
terms of stability. A number of numerical test cases, ranging from analytical
tests to near-realistic laboratory benchmarks, demonstrate the performance of
the method for inundation applications. In particular, super-linear
convergence, mass-conservation, well-balancedness, and stability are verified
Well-balanced -adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations
In this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on -adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solution is observed. Obtaining well-balanced numerical schemes for the shallow water equations on fixed meshes is nontrivial and has been a topic of much research. In [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887–1922] we introduced a well-balanced discontinuous Galerkin method using the theory of weak solutions for nonconservative products introduced in [G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548]. In this article we continue this approach and prove well-balancedness of a discontinuous Galerkin method for the shallow water equations on moving meshes. Numerical simulations are then performed to verify the -adaptive method in combination with the space-time discontinuous Galerkin method against analytical solutions and showing its robustness on more complex problems
An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
We extend the entropy stable high order nodal discontinuous Galerkin spectral
element approximation for the non-linear two dimensional shallow water
equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J.
Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin
method for the two dimensional shallow water equations on unstructured
curvilinear meshes with discontinuous bathymetry. Journal of Computational
Physics, 340:200-242, 2017] with a shock capturing technique and a positivity
preservation capability to handle dry areas. The scheme preserves the entropy
inequality, is well-balanced and works on unstructured, possibly curved,
quadrilateral meshes. For the shock capturing, we introduce an artificial
viscosity to the equations and prove that the numerical scheme remains entropy
stable. We add a positivity preserving limiter to guarantee non-negative water
heights as long as the mean water height is non-negative. We prove that
non-negative mean water heights are guaranteed under a certain additional time
step restriction for the entropy stable numerical interface flux. We implement
the method on GPU architectures using the abstract language OCCA, a unified
approach to multi-threading languages. We show that the entropy stable scheme
is well suited to GPUs as the necessary extra calculations do not negatively
impact the runtime up to reasonably high polynomial degrees (around ). We
provide numerical examples that challenge the shock capturing and positivity
properties of our scheme to verify our theoretical findings
Non-Linear Shallow Water Equations numerical integration on curvilinear boundary-conforming grids
An Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the Shallow Water Equations on generalized curvilinear coordinate systems is proposed. The Shallow Water Equations are expressed in a contravariant formulation in which Christoffel symbols are avoided. The equations are solved by using a high-resolution finite-volume method incorporated with an exact Riemann Solver. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities on generalized boundary-conforming grids is presented; this procedure allows the numerical scheme to satisfy the freestream preservation property on highly-distorted grids. The capacity of the proposed model is verified against test cases present in literature. The results obtained are compared with analytical solutions and alternative numerical solutions
A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers
We present a novel staggered semi-implicit hybrid FV/FE method for the
numerical solution of the shallow water equations at all Froude numbers on
unstructured meshes. A semi-discretization in time of the conservative
Saint-Venant equations with bottom friction terms leads to its decomposition
into a first order hyperbolic subsystem containing the nonlinear convective
term and a second order wave equation for the pressure. For the spatial
discretization of the free surface elevation an unstructured mesh of triangular
simplex elements is considered, whereas a dual grid of the edge-type is
employed for the computation of the depth-averaged momentum vector. The first
stage of the proposed algorithm consists in the solution of the nonlinear
convective subsystem using an explicit Godunov-type FV method on the staggered
grid. Next, a classical continuous FE scheme provides the free surface
elevation at the vertex of the primal mesh. The semi-implicit strategy followed
circumvents the contribution of the surface wave celerity to the CFL-type time
step restriction making the proposed algorithm well-suited for low Froude
number flows. The conservative formulation of the governing equations also
allows the discretization of high Froude number flows with shock waves. As
such, the new hybrid FV/FE scheme is able to deal simultaneously with both,
subcritical as well as supercritical flows. Besides, the algorithm is well
balanced by construction. The accuracy of the overall methodology is studied
numerically and the C-property is proven theoretically and validated via
numerical experiments. The solution of several Riemann problems attests the
robustness of the new method to deal also with flows containing bores and
discontinuities. Finally, a 3D dam break problem over a dry bottom is studied
and our numerical results are successfully compared with numerical reference
solutions and experimental data
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