841 research outputs found
Poisson integrators
An overview of Hamiltonian systems with noncanonical Poisson structures is
given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are
presented. Numerical integrators using generating functions, Hamiltonian
splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems
and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems
and the discrete gradient methods are also presented.Comment: 30 page
Notes on the Discontinuous Galerkin methods for the numerical simulation of hyperbolic equations 1 General Context 1.1 Bibliography
The roots of Discontinuous Galerkin (DG) methods is usually attributed to
Reed and Hills in a paper published in 1973 on the numerical approximation of
the neutron transport equation [18]. In fact, the adventure really started with
a rather thoroughfull series of five papers by Cockburn and Shu in the late
80's [7, 5, 9, 6, 8]. Then, the fame of the method, which could be seen as a
compromise between Finite Elements (the center of the method being a weak
formulation) and Finite Volumes (the basis functions are defined cell-wise, the
cells being the elements of the primal mesh) increased and slowly investigated
successfully all the domains of Partial Differential Equations numerical
integration. In particular, one can cite the ground papers for the common
treatment of convection-diffusion equations [4, 3] or the treatment of pure
elliptic equations [2, 17]. For more information on the history of
Discontinuous Galerkin method, please refer to section 1.1 of [15]. Today, DG
methods are widely used in all kind of manners and have applications in almost
all fields of applied mathematics. (TODO: cite applications and
structured/unstructured meshes, steady/unsteady, etc...). The methods is now
mature enough to deserve entire text books, among which I cite a reference book
on Nodal DG Methods by Henthaven and Warburton [15] with the ground basis of DG
integration, numerical analysis of its linear behavior and generalization to
multiple dimensions. Lately, since 2010, thanks to a ground work of Zhang and
Shu [26, 27, 25, 28, 29], Discontinuous Galerkin methods are eventually able to
combine high order accuracy and certain preservation of convex constraints,
such as the positivity of a given quantity, for example. These new steps
forward are very promising since it brings us very close to the "Ultimate
Conservative Scheme", [23, 1]
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
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
Splitting and composition methods in the numerical integration of differential equations
We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table
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