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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - HNSKMAP (High-order Numerical Schemes for Kinetic Models with Applications in Plasma Physics)

Teaser

Summary

We are interested in the mathematical and numerical analysis of mathematical models coming from kinetic theory. The main application are for plasma physics, semi-conductors, polymers, traffic networking etc.

On the one hand, we want to propose and analyse systematic numerical methods for nonlinear kinetic models which have some challenging difficulties such as physical conservations, asymptotic regimes and stiffness. On the other hand, applications to plasma physics will be investigated, which are mainly high dimensional problems with multi-scale and complex geometries. Moreover collisions between particles for large time scale simulation need to be taken into account.
We would like to develop a class of less dissipative high order Hermite methods together with weighted essentially non-oscillatory (WENO) reconstructions to control spurious numerical oscillations, and high order asymptotic preserving (AP) discontinuous Galerkin (DG) schemes with implicit-explicit (IMEX) time discretizations for multi-scale stiff problems under unresolved meshes. More importantly, these developed numerical methods would satisfy the positivity preserving (PP) principle, such as positive density distribution functions for kinetic descriptions, which is often violated by high order numerical methods with physical meaningless values.

Work performed

Our objectives are design and analysis accurate and stable high order numerical methods which can be efficiently and effectively extended to high dimensions and suitable for parallelizing. The asymptotic preserving property for capturing different scales in different physical regimes, conservation and less dissipation for large time scales, fast algorithms for the global collisional integral operators, would be taken into account. Moreover, high order positivity preserving flux limiters would be incorporated to guarantee positive distribution functions with physical meanings. Finally these developed methods would be applied to plasma physical problems with complex geometries.
More precisely, the first semester is devoted to the modeling part where we have derived a reduced kinetic model starting to the 2Dx3D Vlasov-Maxwell system in the presence of a large external magnetic field. Then the goal is to perform some numerical simulations and to validate our approach on suitable numerical tests. The main problem is that there are really few results in the literature on this topic therefore the numerical validation is crucial here.

Final results

We first consider a typical configuration of tokamak plasma and develop an asymptotic model with low dimension. The plasma is confined by a strong external magnetic field, hence the charged gas evolves under its self-consistent electromagnetic field and the confining magnetic field. We assume that on the time scale we consider, collisions can be neglected both for ions and electrons, hence collective effects are dominant and the plasma is entirely modeled with kinetic Vlasov-Maxwell equations. Such a kinetic model provides an appropriate description of turbulent transport in a fairly general context, but it requires to solve a six dimensional problem which leads to a huge computational cost.
To reduce the cost of numerical simulations, it is classical to derive asymptotic models with a smaller number of variables than the kinetic description.