Projective and telescopic projective integration for the nonlinear BGK\n and Boltzmann equations

Abstract

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