Abstract
This work concerns the numerical simulation of the Vlasov-Poisson equation using semi-Lagrangian methods on Graphics Processing Units (GPU). To accomplish this goal, modifications to traditional methods had to be implemented. First and foremost, a reformulation of semi-Lagrangian methods is performed, which enables us to rewrite the governing equations as a circulant matrix operating on the vector of unknowns. This product calculation can be performed efficiently using FFT routines. Nowadays GPU is no more limited to single precision; however, single precision may still be preferred with respect to performance and available memory. So, in order to be able to deal with single precision, a δf type method is adopted which only needs refinement in specialized areas of phase space but not throughout. Thus, a GPU Vlasov-Poisson solver can indeed perform high precision simulations (since it uses very high order of reconstruction and a large number of grid points in phase space). We show results for more academic test cases and also for physically relevant phenomena such as the bump on tail instability and the simulation of Kinetic Electrostatic Electron Nonlinear (KEEN) waves.
Highlights
At the one body distribution function level, the kinetic theory of charged particles interacting with electrostatic fields and ignoring collisions, may be described by the Vlasov-Poisson system of equations
We propose two slight modifications of the semi-Lagrangian method which enable the use of single precision computations while at the same time recovering the precision reached by a double precision CPU code
Most of the work is on the Fast Fourier Transform (FFT), which is optimized for CUDA in the cufft library, and is transparent for the user
Summary
At the one body distribution function level, the kinetic theory of charged particles interacting with electrostatic fields and ignoring collisions, may be described by the Vlasov-Poisson system of equations. Semi-Lagrangian methods try to retain the best features of the two approaches: the phase space distribution function is updated by solving backward the equations of motion (i.e. the characteristics), and by using an interpolation step to remap the solution onto the phase space grid These methods are often implemented in a split-operator framework. They are very fast enabling us to test and compare different interpolation operators (very high order Lagrangian or spline reconstructions) using a large number of grid points per direction in phase space. These involve several comparisons between the different methods and orders of numerical approximation and their performances on GPU and CPU on three canonical test problems
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