5,980 research outputs found
On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws
In this paper, we present a shock capturing discontinuous Galerkin (SC-DG)
method for nonlinear systems of conservation laws in several space dimensions
and analyze its stability and convergence. The scheme is realized as a
space-time formulation in terms of entropy variables using an entropy stable
numerical flux. While being similar to the method proposed in [14], our
approach is new in that we do not use streamline diffusion (SD) stabilization.
It is proved that an artificial-viscosity-based nonlinear shock capturing
mechanism is sufficient to ensure both entropy stability and entropy
consistency, and consequently we establish convergence to an entropy
measure-valued (emv) solution. The result is valid for general systems and
arbitrary order discontinuous Galerkin method.Comment: Comments: Affiliations added Comments: Numerical results added,
shortened proo
A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws
We present reliable a posteriori estimators for some fully discrete schemes
applied to nonlinear systems of hyperbolic conservation laws in one space
dimension with strictly convex entropy. The schemes are based on a method of
lines approach combining discontinuous Galerkin spatial discretization with
single- or multi-step methods in time. The construction of the estimators
requires a reconstruction in time for which we present a very general framework
first for odes and then apply the approach to conservation laws. The
reconstruction does not depend on the actual method used for evolving the
solution in time. Most importantly it covers in addition to implicit methods
also the wide range of explicit methods typically used to solve conservation
laws. For the spatial discretization, we allow for standard choices of
numerical fluxes. We use reconstructions of the discrete solution together with
the relative entropy stability framework, which leads to error control in the
case of smooth solutions. We study under which conditions on the numerical flux
the estimate is of optimal order pre-shock. While the estimator we derive is
computable and valid post-shock for fixed meshsize, it will blow up as the
meshsize tends to zero. This is due to a breakdown of the relative entropy
framework when discontinuities develop. We conclude with some numerical
benchmarking to test the robustness of the derived estimator
Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms
We consider nonlinear hyperbolic conservation laws, posed on a differential
(n+1)-manifold with boundary referred to as a spacetime, and in which the
"flux" is defined as a flux field of n-forms depending on a parameter (the
unknown variable). We introduce a formulation of the initial and boundary value
problem which is geometric in nature and is more natural than the vector field
approach recently developed for Riemannian manifolds. Our main assumption on
the manifold and the flux field is a global hyperbolicity condition, which
provides a global time-orientation as is standard in Lorentzian geometry and
general relativity. Assuming that the manifold admits a foliation by compact
slices, we establish the existence of a semi-group of entropy solutions.
Moreover, given any two hypersurfaces with one lying in the future of the
other, we establish a "contraction" property which compares two entropy
solutions, in a (geometrically natural) distance equivalent to the L1 distance.
To carry out the proofs, we rely on a new version of the finite volume method,
which only requires the knowledge of the given n-volume form structure on the
(n+1)-manifold and involves the {\sl total flux} across faces of the elements
of the triangulations, only, rather than the product of a numerical flux times
the measure of that face.Comment: 26 page
On the Convergence of Space-Time Discontinuous Galerkin Schemes for Scalar Conservation Laws
We prove convergence of a class of space-time discontinuous Galerkin schemes
for scalar hyperbolic conservation laws. Convergence to the unique entropy
solution is shown for all orders of polynomial approximation, provided strictly
monotone flux functions and a suitable shock-capturing operator are used. The
main improvement, compared to previously published results of similar scope, is
that no streamline-diffusion stabilization is used. This is the way
discontinuous Galerkin schemes were originally proposed, and are most often
used in practice
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