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Ward Melis; Thomas Rey; Giovanni Samaey
Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations
SMAI-Journal of computational mathematics, 5 (2019), p. 53-88, doi: 10.5802/smai-jcm.43
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Class. Math.: 82B40, 76P05, 65M70, 65M08, 65M12
Keywords: Boltzmann equation, BGK equation, Projective Integration, spectral theory, fast spectral scheme


We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.


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