1,429 research outputs found
Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws
In this paper we analyze the large time asymptotic behavior of the discrete
solutions of numerical approximation schemes for scalar hyperbolic conservation
laws. We consider three monotone conservative schemes that are consistent with
the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and
Godunov. We mainly focus on the inviscid Burgers equation, for which we know
that the large time behavior is of self-similar nature, described by a
two-parameter family of N-waves. We prove that, at the numerical level, the
large time dynamics depends on the amount of numerical viscosity introduced by
the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic
behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles,
corresponding to the asymptotic behavior of the solutions of the continuous
viscous Burgers equation. The same problem is analyzed in the context of
self-similar variables that lead to a better numerical performance but to the
same dichotomy on the asymptotic behavior: N-waves versus viscous ones. We also
give some hints to extend the results to more general fluxes. Some numerical
experiments illustrating the accuracy of the results of the paper are also
presented.Comment: Error corrected in main theorem in v3. Obtained results do not change
in essence. Minor typos corrected in v
On the upstream mobility scheme for two-phase flow in porous media
When neglecting capillarity, two-phase incompressible flow in porous media is
modelled as a scalar nonlinear hyperbolic conservation law. A change in the
rock type results in a change of the flux function. Discretizing in
one-dimensional with a finite volume method, we investigate two numerical
fluxes, an extension of the Godunov flux and the upstream mobility flux, the
latter being widely used in hydrogeology and petroleum engineering. Then, in
the case of a changing rock type, one can give examples when the upstream
mobility flux does not give the right answer.Comment: A preprint to be published in Computational Geoscience
A moving mesh method for one-dimensional hyperbolic conservation laws
We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work
The discrete one-sided Lipschitz condition for convex scalar conservation laws
Physical solutions to convex scalar conservation laws satisfy a one-sided Lipschitz condition (OSLC) that enforces both the entropy condition and their variation boundedness. Consistency with this condition is therefore desirable for a numerical scheme and was proved for both the Godunov and the Lax-Friedrichs scheme--also, in a weakened version, for the Roe scheme, all of them being only first order accurate. A new, fully second order scheme is introduced here, which is consistent with the OSLC. The modified equation is considered and shows interesting features. Another second order scheme is then considered and numerical results are discussed
An Unsplit, Cell-Centered Godunov Method for Ideal MHD
We present a second-order Godunov algorithm for multidimensional, ideal MHD.
Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys.
vol. 87, 1990), with all of the primary dependent variables centered at the
same location. To properly represent the divergence-free condition of the
magnetic fields, we apply a discrete projection to the intermediate values of
the field at cell faces, and apply a filter to the primary dependent variables
at the end of each time step. We test the method against a suite of linear and
nonlinear tests to ascertain accuracy and stability of the scheme under a
variety of conditions. The test suite includes rotated planar linear waves, MHD
shock tube problems, low-beta flux tubes, and a magnetized rotor problem. For
all of these cases, we observe that the algorithm is second-order accurate for
smooth solutions, converges to the correct weak solution for problems involving
shocks, and exhibits no evidence of instability or loss of accuracy due to the
possible presence of non-solenoidal fields.Comment: 37 Pages, 9 Figures, submitted to Journal of Computational Physic
Local characteristic algorithms for relativistic hydrodynamics
Numerical schemes for the general relativistic hydrodynamic equations are
discussed. The use of conservative algorithms based upon the characteristic
structure of those equations, developed during the last decade building on
ideas first applied in Newtonian hydrodynamics, provides a robust methodology
to obtain stable and accurate solutions even in the presence of
discontinuities. The knowledge of the wave structure of the above system is
essential in the construction of the so-called linearized Riemann solvers, a
class of numerical schemes specifically designed to solve nonlinear hyperbolic
systems of conservation laws. In the last part of the review some astrophysical
applications of such schemes, using the coupled system of the
(characteristic) Einstein and hydrodynamic equations, are also briefly
presented.Comment: 20 pages, 4 figures, To appear in the proceedings of the workshop
"The conformal structure of space-time", J. Frauendiener, H. Friedrich, eds,
Springer Lecture Notes in Physic
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