We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on ...

Matrix Riccati Differential Equations (MRDEs) are initial value problems of the form: X 0 1⁄4 A21 XA11 þ A22X XA12X; Xð0Þ 1⁄4 X0: These equations arise frequently throughout applied mathematics, science, and engineering. It can happen that even when the Aij are smooth functions of t or constant, the solution X may have a singularity or even infinitely many singularities. This paper shows severa...

We investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation (AD). By differentiating the time step controller and the error estimator of the original algorithm, reverse mode AD generates spurious adjoint derivatives of the time steps. The resulting discrete adjoint models become inconsistent with the adjoint ODE, and yield incor...

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several firstorder Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious soluti...