405 research outputs found
Fully discrete hyperbolic initial boundary value problems with nonzero initial data
The stability theory for hyperbolic initial boundary value problems relies
most of the time on the Laplace transform with respect to the time variable.
For technical reasons, this usually restricts the validity of stability
estimates to the case of zero initial data. In this article, we consider the
class of non-glancing finite difference approximations to the hyperbolic
operator. We show that the maximal stability estimates that are known for zero
initial data and nonzero boundary source term extend to the case of nonzero
initial data in \^a 2 . The main novelty of our approach is to cover finite
difference schemes with an arbitrary number of time levels. As an easy
corollary of our main trace estimate, we recover former stability results in
the semigroup sense by Kreiss [Kre68] and Osher [Osh69b]
The Leray- GĂĄrding method for finite difference schemes
International audienceIn [Ler53] and [ GĂĄr56], Leray and GĂĄrding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in [Ler53, GĂĄr56 ] provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multiplier is the starting point of the argument by Rauch in [Rau72] for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels, and encompasses a somehow magical trick that has been known for a long time for the leapfrog scheme. More importantly, the existence and properties of the local multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems, which answers a problem raised by Trefethen, Kreiss and Wu [Tre84, KW93]
Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach
The long-term distributions of trajectories of a flow are described by
invariant densities, i.e. fixed points of an associated transfer operator. In
addition, global slowly mixing structures, such as almost-invariant sets, which
partition phase space into regions that are almost dynamically disconnected,
can also be identified by certain eigenfunctions of this operator. Indeed,
these structures are often hard to obtain by brute-force trajectory-based
analyses. In a wide variety of applications, transfer operators have proven to
be very efficient tools for an analysis of the global behavior of a dynamical
system.
The computationally most expensive step in the construction of an approximate
transfer operator is the numerical integration of many short term trajectories.
In this paper, we propose to directly work with the infinitesimal generator
instead of the operator, completely avoiding trajectory integration. We propose
two different discretization schemes; a cell based discretization and a
spectral collocation approach. Convergence can be shown in certain
circumstances. We demonstrate numerically that our approach is much more
efficient than the operator approach, sometimes by several orders of magnitude
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
Operator splittings and spatial approximations for evolution equations
The convergence of various operator splitting procedures, such as the
sequential, the Strang and the weighted splitting, is investigated in the
presence of a spatial approximation. To this end a variant of Chernoff's
product formula is proved. The methods are applied to abstract partial delay
differential equations.Comment: to appear in J. Evol. Equations. Reviewers comments are incorporate
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