358 research outputs found
On discretely entropy conservative and entropy stable discontinuous Galerkin methods
High order methods based on diagonal-norm summation by parts operators can be
shown to satisfy a discrete conservation or dissipation of entropy for
nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as
nodal discontinuous Galerkin methods with diagonal mass matrices. In this work,
we describe how use flux differencing, quadrature-based projections, and
SBP-like operators to construct discretely entropy conservative schemes for DG
methods under more arbitrary choices of volume and surface quadrature rules.
The resulting methods are semi-discretely entropy conservative or entropy
stable with respect to the volume quadrature rule used. Numerical experiments
confirm the stability and high order accuracy of the proposed methods for the
compressible Euler equations in one and two dimensions
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]
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