On positivity of the local iteration modified time integration scheme

M. A. Botchev, V. T. Zhukov


Abstract: Local iteration modified (LIM) scheme is an explicit time integration method where accuracy and stability for large time steps are provided by Chebyshev polynomial iterations. The LIM scheme possesses two characteristic features: (1) the Chebyshev polynomial in the LIM scheme is constructed to give a higher accuracy for the important low frequency solution modes, (2) the number of Chebyshev iterations is chosen depending on time step size, to have a stable time integration (rather than to solve arising linear system as done in implicit schemes). An important property of the LIM scheme which makes it unique in the class of explicit iterative time integration schemes, is positivity. In this work we thoroughly study this property of the scheme. Up to now, positivity of the LIM scheme has been shown experimentally in numerical tests as well as, partially, theoretically. Our aim here is to provide further theoretical and practical insight in the positivity properties of the scheme.

Keywords: local iteration modified (LIM) scheme, positivity, monotonicity, explicit stabilized Runge–Kutta schemes, Runge–Kutta–Chebyshev methods, exponential time integration.