A semi-Lagrangian Vlasov solver in tensor train format

In this article, we derive a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Grid-based methods for the Vlasov equation have been shown to give accurate results but their use has mostly been limited to simulations in two dimensional phase space due to extensive memory requirements in higher dimensions. Compression of the solution via high-order singular value decomposition can help in reducing the storage requirements and the tensor train (TT) format provides efficient basic linear algebra routines for low-rank representations of tensors. In this paper, we develop interpolation formulas for a semi-Lagrangian solver in TT format. In order to efficiently implement the method, we propose a compression of the matrix representing the interpolation step and an efficient implementation of the Hadamard product. We show numerical simulations for standard test cases in two, four and six dimensional phase space. Depending on the test case, the memory requirements reduce by a factor $10^2-10^3$ in four and a factor $10^5-10^6$ in six dimensions compared to the full-grid method

The semi-Lagrangian method on curvilinear grids

International audienceWe study the semi-Lagrangian method on curvilinear grids. The classical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time

Gyrokinetic Vlasov-Poisson model derived by hybrid-coordinate transform of the distribution function

This paper points out that the full-orbit density obtained in the standard elec-trostatic gyrokinetic model is not truly accurate at the order ε σ−1 with respect to the equilibrium distribution e −αµ with µ ∈ (0, µ max), where ε is the order of the normalized Larmor radius, ε σ the order of the amplitude of the normalized elec-trostatic potential, and α a factor of O(1). This error makes the exact order of the full-orbit density not consistent with that of the approximation of the full-orbit distribution function. By implementing a hybrid coordinate frame to get the full-orbit distribution, specifically, by replacing the magnetic moment on the full-orbit coordinate frame with the one on the gyrocenter coordinate frame to derive the full-orbit distribution transformed from the gyrocenter distribution, it's proved that the full-orbit density can be approximated with the exact order being ε σ−1. The numerical comparison between the new gyrokinetic model and the standard one was carried out using Selalib code for an initial distribution proportional to exp(−µB T i) in constant cylindrical magnetic field configuration with the existence of electro-static perturbations. The simulation results show that the saturation time of the new mode is later than that of the standard one and there exists obvious difference of the evolution of the amplitude of the radial short-scale potential perturbations between the two models