We present the design and implementation of an L 2 -stable spectral method for the discretization of the Vlasov-Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank-Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton-Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre-Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L 2 -stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.
We demonstrate the improvements to an implicit Particle-in-Cell code, iPic3D, on the example of dipolar magnetic field immersed in the flow of the plasma and show the formation of a magnetosphere. We address the problem of modelling multi-scale phenomena during the formation of a magnetosphere by implementing an adaptive sub-cycling technique to resolve the motion of particles located close to the magnetic dipole centre, where the magnetic field intensity is maximum. In addition, we implemented new open boundary conditions to model the inflow and outflow of plasma. We present the results of a global three-dimensional Particle-in-Cell simulation and discuss the performance improvements from the adaptive sub-cycling technique.
A spectral method for kinetic plasma simulations based on the expansion of the velocity distribution function in a variable number of Hermite polynomials is presented. The method is based on a set of non-linear equations that is solved to determine the coefficients of the Hermite expansion satisfying the Vlasov and Poisson equations. In this paper, we first show that this technique combines the fluid and kinetic approaches into one framework. Second, we present an adaptive strategy to increase and decrease the number of Hermite functions dynamically during the simulation. The technique is applied to the Landau damping and two-stream instability test problems. Performance results show 21% and 47% saving of total simulation time in the Landau and two-stream instability test cases, respectively.
The vast majority of parallel scientific applications distributes computation among processes that are in a busy state when computing and in an idle state when waiting for information from other processes. We identify the propagation of idle waves through processes in scientific applications with a local information exchange between the two processes. Idle waves are nondispersive and have a phase velocity inversely proportional to the average busy time. The physical mechanism enabling the propagation of idle waves is the local synchronization between two processes due to remote data dependency. This study provides a description of the large number of processes in parallel scientific applications as a continuous medium. This work also is a step towards an understanding of how localized idle periods can affect remote processes, leading to the degradation of global performance in parallel scientific applications.
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