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PhD Thesis of Eric Madaule: development of adaptive schemes for VlasovPoisson: GalWa
Author of the codes: E. Madaule
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Article:
PhD Thesis manuscript (in French) of Eric Madaule (Supervisor: Nicolas Besse; Cosupervisor: Erwan Deriaz)
Figure 1: Damping of electric field for the weak Landau damping case.
Figure 2: Mesh at t=80 plasma periods for bump on tail.
Figure 3: Distribution at t=80 plasma periods for bump on tail.
The purpose of my PhD was to develop and compare different new numerical schemes involving adaptive mesh refinement for the resolution of the VlasovPoisson problem. VlasovPoisson equations are used both in plasma physic and in astrophysics. The main difference between the two contexts is the sign of the Poisson equation but this has no impact on the computation scheme. We implement Adaptive Mesh Refinement part (AMR) using multiwavelet decomposition as in Alpert et al. (2002) and Alpert (1993). The method can be very easily coupled with a discontinuous Galerkin approach for the space discretisation since both parts use highorder local polynomial reconstruction. Time resolution is performed by combining the use of semiLagrangian time integration and Strang splitting technique (order 2). The first semiLagrangian scheme, that we call semiLagrangian discontinuous Galerkin (SLDG), was developed in Qui and Shu (2011). In this approach, we transform a time integration into a space integration along the characteristic curves. The second semiLagrangian scheme, that we call characteristic discontinuous Galerkin (CDG), was first proposed by Pironneau (1982) and Douglas & Russel (1982), renewed by Childs & Morton (1990) and recently by Guo et al. (2014). This approach computes the foot of the characteristics arriving at the boundaries of the cell at t^{n+1} to get the starting cell and applies the coefficients from this starting cell to the polynomials on the arrival cell. This simply corresponds to cell transportation. Adaptivity is performed in one dimension with algorithms from Alpert et al. (2002) and Alpert (1993) and is extended to higher number of dimensions through tensorial products. All these aspects have been tested on translation and rotation cases and show good conservation of numerical mass.
I ran several test cases for 2D VlasovPoisson in plasma physics such as weak and strong Landau damping, twostream instability and bump on tail, as well as test cases for astrophysics, such as a cold layer or Jeans instability. In all cases, we see good conservation of numerical mass. L^{1}, L^{2} norms, extrema are not exactly conserved but their relative variation remains very small. However, such conservation violations are expected and consistent with other semiLagrangian schemes. The decrease of electric field for weak Landau damping is shown on Figure 1 and is in good agreement with analytical predictions. As one can see, SLDG and CDG give almost the same results (differences can be seen with very close attention paid to data files). Figures 2 and 3 respectively show mesh and distribution at t=80 plasma periods for bump on tail. These are obtained for polynomials of degree 2 (in 1D, so 9 variables per cell) with 8 levels of refinement (up to 256 cells per direction in full domain). The mesh follows very well the details of the solution.
In order to take full advantage of adaptivity future developments include the resolution of the relativistic Vlasovpoisson problem presented by Arber, Dendy & Sircombe (2005), involving ions and electrons and on some relativistic VlasovMaxwell problems found in Besse et al. (2008). I also started the implementation of the four dimensional case.
Bibliography
 A class of Bases in L2 for the Sparse Representation of Integral Operator, B. K. Alpert, SIAM Journal on Mathematical Analysis 24.1, 246262, 1993
 Adaptive Solution of Partial Differential Equations in Multiwavelet Bases, B. Alpert, G. Beylkin, D. Gines, L. Vozovoi, JCP 182, 149190, 2002
 Accelerated electron population formed by Langmuir wavecaviton interactions, T. D. Arber, R. O. Dendy, N. J. Sircombe, Physics of Plasma, 2005
 A waveletMRAbased adaptive semiLagrangian method for the relativistic VlasovMaxwell system, N. Besse, G. Latu, P. Bertrand, A Ghizzo, E. Sonnendrücker, Journal of Computational Physics 227, 78897916, 2008
 Characteristic Galerkin methods for scalar conservation law in one dimension, P. N. Childs, K. W. Morton, SIAM J. Numer. Anal. 27, 553594, 1990
 Numerical methods for convectiondiffusion problems based on combining the method of characteristics with finite element or finite difference procedures, J. Douglas, T.F. Russel, SIAM J. Numer. Anal. 19, 871885, 1982
 On the transportdiffusion algorithm and its application to the NavierStokes equations, O. Pironneau, Numer. Math. 38, 309332, 1982
 Positivity preserving semiLagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the VlasovPoisson system, JingMei Qiu, ChiWang Shu, Journal of Computational Physics 230, 83868409, 2011
 A conservative semiLagrangian discontinuous Galerkin scheme on the cubedsphere, Wei Guo, Ramachandran D. Nair, JingMei Qiu, Mon. Wea. Rev., 142, 457475, 2014
