An upwind LDG scheme for the Fokker-Planck-Landau equation that conserves mass, momentum and a projected energy by building the discrete collision kernel from discrete gradients of the projected energy.
Convergence and Error Estimates for the Conservative Spectral Method for Fokker-Planck-Landau Equations
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abstract
Error estimates are rigorously derived for a semi-discrete version of a conservative spectral method for approximating the space-homogeneous Fokker-Planck-Landau (FPL) equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in $L^2$ spaces globally time, under certain conditions. These estimates, combined with control of the spectral projection, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates FPL equations associated to any range of potentials.
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A structure-preserving local discontinuous Galerkin method for the Fokker-Planck-Landau equation
An upwind LDG scheme for the Fokker-Planck-Landau equation that conserves mass, momentum and a projected energy by building the discrete collision kernel from discrete gradients of the projected energy.