Séminaire Calcul Scientifique et Modélisation

On the mesoscopic level, motion of individual particles can be modeled by a kinetic transport equation for the population density f(t,x,v) as a function of time t, space x and velocity v \in V. A relaxation term on the right hand side accounts for scattering due to self-induced velocity changes and typically involves a parameter K(x,v,v') encoding the probability of changing from velocity v' to v at location x:

\partial_t f(t,x,v) + v \cdot abla f(t,x,v) = \int K(x,v,v') f(t,x,v') - K(x,v',v)f(t,x,v) dv'

This hyperbolic model is widely used to model bacterial motion, called chemotaxis.

We study the inverse parameter reconstruction problem whose aim is to recover the scattering parameter $K$ and that has to be solved when fitting the model to a real situation. We restrict ourselves to macroscopic, i.e. velocity averaged data $\rho = \int f dv$ as a basis of our reconstruction. This introduces additional difficulties, which can be overcome by the use of short time interior domain data. In this way, we can establish theoretical existence and uniqueness of the reconstruction, study its macroscopic limiting behavior and numerically conduct the inversion under suitable data generating experimental designs.

This work based on a collaboration with Kathrin Hellmuth (Würzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).

The efficient numerical solution of compressible Euler equations of gas dynamics poses several computational challenges. Severe time restrictions are required by standard explicit time discretization techniques, in particular for flow regimes characterized by low Mach number values. We analyze schemes based on a general Implicit-Explicit (IMEX) Runge-Kutta (RK) time discretization for the compressible Euler equations of gas dynamics. We prove the asymptotic-preserving (AP) property of the numerical scheme in the low Mach number limit for both a single asymptotic length scale and two length scales. The analysis is carried out considering a general equation of state (EOS) and therefore it is not restricted to the ideal gas law as done for standard AP schemes. We couple implicitly the energy equation to the momentum one, while treating the continuity equation in an explicit fashion.

We also present an alternative strategy, which consists of employing a semi-implicit temporal integrator based on IMEX-RK methods (SI-IMEX-RK). The stiff dependence is carefully analyzed, so as to avoid the solution of a nonlinear equation for the pressure also for equations of state (EOS) of non-ideal gases.

The asymptotic-preserving (AP) and the asymptotically-accurate (AA) properties of the two approaches are assessed on a number of classical benchmarks for ideal gases and on their extension to non-ideal gases. Particular attention is devoted to the robustness of the numerical methods with respect to the boundary conditions and to non well-prepared initial conditions.

This work is based on a collaboration with Luca Bonaventura (Politecnico di Milano), Sebastiano Boscarino and Giovanni Russo (Università di Catania).