16,259 research outputs found
Coupling techniques for nonlinear hyperbolic equations. IV. Multi-component coupling and multidimensional well-balanced schemes
This series of papers is devoted to the formulation and the approximation of
coupling problems for nonlinear hyperbolic equations. The coupling across an
interface in the physical space is formulated in term of an augmented system of
partial differential equations. In an earlier work, this strategy allowed us to
develop a regularization method based on a thick interface model in one space
variable. In the present paper, we significantly extend this framework and, in
addition, encompass equations in several space variables. This new formulation
includes the coupling of several distinct conservation laws and allows for a
possible covering in space. Our main contributions are, on one hand, the design
and analysis of a well-balanced finite volume method on general triangulations
and, on the other hand, a proof of convergence of this method toward entropy
solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a
single conservation law without coupling). The core of our analysis is, first,
the derivation of entropy inequalities as well as a discrete entropy
dissipation estimate and, second, a proof of convergence toward the entropy
solution of the coupling problem.Comment: 37 page
Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation
In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [1] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.
The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.
[1] R. Hiptmair. Multigrid method for Maxwell's equations. SIAM J. Numer. Anal., 36(1):204-225, 1999
A Toy Model for Testing Finite Element Methods to Simulate Extreme-Mass-Ratio Binary Systems
Extreme mass ratio binary systems, binaries involving stellar mass objects
orbiting massive black holes, are considered to be a primary source of
gravitational radiation to be detected by the space-based interferometer LISA.
The numerical modelling of these binary systems is extremely challenging
because the scales involved expand over several orders of magnitude. One needs
to handle large wavelength scales comparable to the size of the massive black
hole and, at the same time, to resolve the scales in the vicinity of the small
companion where radiation reaction effects play a crucial role. Adaptive finite
element methods, in which quantitative control of errors is achieved
automatically by finite element mesh adaptivity based on posteriori error
estimation, are a natural choice that has great potential for achieving the
high level of adaptivity required in these simulations. To demonstrate this, we
present the results of simulations of a toy model, consisting of a point-like
source orbiting a black hole under the action of a scalar gravitational field.Comment: 29 pages, 37 figures. RevTeX 4.0. Minor changes to match the
published versio
Harmonic Initial-Boundary Evolution in General Relativity
Computational techniques which establish the stability of an
evolution-boundary algorithm for a model wave equation with shift are
incorporated into a well-posed version of the initial-boundary value problem
for gravitational theory in harmonic coordinates. The resulting algorithm is
implemented as a 3-dimensional numerical code which we demonstrate to provide
stable, convergent Cauchy evolution in gauge wave and shifted gauge wave
testbeds. Code performance is compared for Dirichlet, Neumann and Sommerfeld
boundary conditions and for boundary conditions which explicitly incorporate
constraint preservation. The results are used to assess strategies for
obtaining physically realistic boundary data by means of Cauchy-characteristic
matching.Comment: 31 pages, 14 figures, submitted to Physical Review
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
Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary
We study nonlinear hyperbolic conservation laws posed on a differential
(n+1)-manifold with boundary referred to as a spacetime, and defined from a
prescribed flux field of n-forms depending on a parameter (the unknown
variable), a class of equations proposed by LeFloch and Okutmustur in 2008. Our
main result is a proof of the convergence of the finite volume method for weak
solutions satisfying suitable entropy inequalities. A main difference with
previous work is that we allow for slices with a boundary and, in addition,
introduce a new formulation of the finite volume method involving the notion of
total flux functions. Under a natural global hyperbolicity condition on the
flux field and the spacetime and by assuming that the spacetime admits a
foliation by compact slices with boundary, we establish an existence and
uniqueness theory for the initial and boundary value problem, and we prove a
contraction property in a geometrically natural L1-type distance.Comment: 32 page
-boundedness of DG()-solutions for nonlinear conservation laws with boundary conditions
We prove the -boundedness of a higher-order
shock-capturing streamline-diffusion DG-method based on polynomials of degree
for general scalar conservation laws. The estimate is given for the
case of several space dimensions and for conservation laws with initial and
boundary conditions
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