33,755 research outputs found
Viscous Boundary Value Problems for Symmetric Systems with Variable Multiplicities
Extending investigations of M\'etivier&Zumbrun in the hyperbolic case, we
treat stability of viscous shock and boundary layers for viscous perturbations
of multidimensional hyperbolic systems with characteristics of variable
multiplicity, specifically the construction of symmetrizers in the
low-frequency regime where variable multiplicity plays a role. At the same
time, we extend the boundary-layer theory to ``real'' or partially parabolic
viscosities, Neumann or mixed-type parabolic boundary conditions, and systems
with nonconservative form, in addition proving a more fundamental version of
the Zumbrun--Serre--Rousset theorem, valid for variable multiplicities,
characterizing the limiting hyperbolic system and boundary conditions as a
nonsingular limit of a reduced viscous system. The new effects of viscosity are
seen to be surprisingly subtle; in particular, viscous coupling of crossing
hyperbolic modes may induce a destabilizing effect. We illustrate the theory
with applications to magnetohydrodynamics
Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation
We consider the development of high order space and time numerical methods
based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic
systems with relaxation. More specifically, we consider hyperbolic balance laws
in which the convection and the source term may have very different time and
space scales. As a consequence the nature of the asymptotic limit changes
completely, passing from a hyperbolic to a parabolic system. From the
computational point of view, standard numerical methods designed for the
fluid-dynamic scaling of hyperbolic systems with relaxation present several
drawbacks and typically lose efficiency in describing the parabolic limit
regime. In this work, in the context of Implicit-Explicit linear multistep
methods we construct high order space-time discretizations which are able to
handle all the different scales and to capture the correct asymptotic behavior,
independently from its nature, without time step restrictions imposed by the
fast scales. Several numerical examples confirm the theoretical analysis
A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation
In this paper we consider the development of Implicit-Explicit (IMEX)
Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such
systems the scaling depends on an additional parameter which modifies the
nature of the asymptotic behavior which can be either hyperbolic or parabolic.
Because of the multiple scalings, standard IMEX Runge-Kutta methods for
hyperbolic systems with relaxation loose their efficiency and a different
approach should be adopted to guarantee asymptotic preservation in stiff
regimes. We show that the proposed approach is capable to capture the correct
asymptotic limit of the system independently of the scaling used. Several
numerical examples confirm our theoretical analysis
On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach
The coupling of hyperbolic and parabolic systems is discussed in a domain Omega divided into two distinct subdomains omega(+) and omega(-). The main concern is to find the proper interface conditions to be fulfilled at the surface separating the two domains. Next, they are used in the numerical approximation of the problem. The justification of the interface conditions is based on a singular perturbation analysis, i.e., the hyperbolic system is rendered parabolic by adding a small artifical viscosity. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the original one, yielding some conditions at the interface. These are taken as interface conditions for the hyperbolic-parabolic problem. Actually, two alternative sets of interface conditions are discussed according to whether the regularization procedure is variational or nonvariational. It is shown how these conditions can be used in the frame of a numerical approximation to the given problem. Furthermore, a method of resolution is discussed which alternates the resolution of the hyperbolic problem within omega(-) and of the parabolic one within omega(+). The spectral collocation method is proposed, as an example of space discretization (different methods could be used as well); both explicit and implicit time-advancing schemes are considered. The present study is a preliminary step toward the analysis of the coupling between Euler and Navier-Stokes equations for compressible flows
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