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## Speaker

Giovanni Russo, Dipartimento di Matematica ed Informatica Università di Catania

## Title

Asymptotic preserving schemes for hyperbolic systems with relaxation

## Abstract

Several physical systems are described by evolutionary partial or integro-partial differential equations. Among them, kinetic models (Boltzmann-type equations, Vlasov-like equation, neutron transport) and models of wave propagation in continuous media (Euler and Navier-Stokes equations in fluid dynamics, elasticity in solid mechanics, magnetofluid dynamics, Saint-Venant model of shallow water) play a relevant role in several applications. From a mathematical viewpoint, many such systems have the structure of hyperbolic systems of balance laws, that must be solved numerically, typically adopting shock-capturing schemes based on finite volume, finite difference, or discontinuous Galerkin space discretization combined with Runge-Kutta of multi-step methods as time integrators [11].

In some cases, such systems contain stiff relaxation terms, which require implicit treatment to avoid excessively small time-steps. In the limit of infinite stiffness, the system reduces to a simplified model involving a smaller number of unknowns. The prototype of such system is the relaxation model by Jin and Xin [10], consisting of a semilinear hyperbolic system of partial differential equations with a stiff relaxation on the right-hand side, which depends on a small parameter * ε*:

As * ε* vanishes, the system formally relaxes to the scalar quasilinear conservation equation * u _{t}* +

*f*(

*u*)

*= 0, while the variable*

_{x}*v*(algebraic variable) is linked to

*u*(differential variable) by the relation

*v*=

*f*(

*u*). This is called hyperbolic to hyperbolic relaxation. The formal convergence becomes rigorous if the so called sub- characteristic condition is satisfied.

If one is interested in the long-time behavior of the relaxation system, then time has to be suitably rescaled. In this case, the limit for vanishing epsilon is a convection-diffusion equation (hyperbolic to parabolic relaxation).

Several relaxation systems have a similar structure, and the condition for the construction of AP schemes that are consistent with the relaxed system in the limit have been widely studied in the literature, both for the hyperbolic to hyperbolic [12], hyperbolic to parabolic [5, 3] and multiscale [4] relaxation. Depending on the property of the relaxation operator, some system with a given number * M* of equations may relax to a balance system with a smaller number * m < M* of equations, which generalizes what is usually studied in the literature, where the relaxed problem is a system of conservation laws.

An effective way to treat these problems is based on Implicit-Explicit (IMEX) schemes, in which the non- stiff hyperbolic part is treated by an explicit scheme, while the stiff relaxation term is treated implicitly [1, 8, 12]. IMEX schemes, coupled with high-order conservative finite difference methods based on non-oscillatory reconstructions and flux-splitting, allow the construction of accurate and efficient schemes which have the property that, in the case of infinite stiffness, they become a consistent discretization of the underlying relaxed system. We say that such schemes are Asymptotic Preserving (AP) [9].

In many cases, systems of balance laws admit stationary solutions, in which flux gradients are balanced by the source term. When small perturbations describing a deviation from the stationary solution are propagated, the signal may be very small, and if the numerical method is not able to preserve stationary solutions exactly or to a great accuracy, then the discretization errors introduced on the stationary solution may severely degrade the accuracy of the signal to be computed. To overcome such a problem, methods that preserve stationary solutions, called Well Balanced (WB), have been developed [2, 6, 7].

The purpose of this talk is to give a brief overview of AP schemes for hyperbolic systems with hyperbolic and parabolic relaxation. The possibility of constructing asymptotic preserving well-balanced schemes will be mentioned.

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