**Topic/Type**:
2.5 Adaptative & multi-scale methods, Oral

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Giovanni Lapenta ^{1}, Stefano Markidis^{2}
**

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^{1} CPA and LMCC, Departement Wiskunde, KU Leuven, Belgium^{2} Nuclear, Plasma, and Radiological Engineering,University of Illinois at Urbana-Champaign, Illinois, USA
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The numerical solution of the kinetic equation of plasma on large physical domains, over long time periods, is one of the biggest challenges in computational physics. The reason is that the numerics itself imposes the use of short time step, fraction of the fundamental oscillation of the plasma(plasma and cyclotron oscillations), and the use of very small grid spacing (~Debye

length), to avoid the uncontrolled growth of spurious numerical modes. However the phenomena of interest for most applications develop on very large scale (thousands of Debye lengths) and over very long time period(millions of fundamental oscillations). A numerical technique to avoid the limitations of kinetic simulations and current effort towards fully kinetic simulations of large-scale

problems, are presented.

Computer simulations of a plasma are inherently multi-scale because they require to study phenomena, spanning over a very large time-scale and length-scale range. The presence of electrons, protons and heavier plasma species with different masses and the multitude of different collective modes and instabilities, cause the presence of multiple scales.

Two main approaches are used for plasma computer simulation: the fluid and the kinetic approaches. Fluid methods, such as Magnetohydrodynamics (MHD) method, capture the macroscopic evolution of the system, but they become inaccurate for problem in which the detailed kinetic processes(wave-particle interactions, wave trapping phenomena) affect the macroscopic behavior of the plasma. The second approach is kinetic, and it aims at knowing the behavior of plasma using the kinetic equation of plasmas. This approach accurately models the plasma from first principles but it is computationally expensive. Often in plasmas, the macroscopic evolution that develops relatively slowly, is strongly coupled with smaller and faster phenomena where kinetic effect are predominant. For instance, the topological changes of magnetic field configuration on large scales, such as the magnetic reconnection, are not possible without kinetic mechanisms at small scales. For these reasons, to retain the kinetic effects is very important in order to describe correctly the overall evolution of the system. The implicit Particle-in-Cell (PIC) is based on a fully kinetic approach but it removes the need to resolve small time scales (plasma frequencies) without eliminating them. The unresolved scales are kept in an approximate way allowing the coupling with slower scales that are fully resolved.

The present contribution is divided in two parts: in the first part, a kinetic numerical method (implicit Particle-in-Cell) for plasma simulations, characterized by large domain, long time range and the use of real plasma parameters is shown. The implicit Particle-in-Cell method, its mathematical derivation, its stability and accuracy properties, and its parallel implementation in petascale computers, are discussed. The main application considered is the kinetic simulation of space weather events.