arXiv subjects

Gustaf Söderlind

Publications and source records attributed to Gustaf Söderlind.

3 recordsLinked to original sources

arXiv2026

Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge-Kutta Methods

In this paper, we investigate how adaptive time-integration strategies can be effectively combined with parallel-in-time numerical methods for solving systems of ordinary differential equations. Our focus is particularly on their influence on tolerance proportionality. We examine various grid-refinement strategies within the multigrid reduction-in-time (MGRIT) framework. Our results show that a simple adjustment to the original refinement factor can substantially improve computational stability and reliability. Through numerical experiments on standard test problems using the XBraid library, we demonstrate that parallel-in-time solutions closely match their sequential counterparts. Moreover, with the use of multiple processors, computing time can be significantly reduced.

math.NA↗

arXiv2018

On the zero-stability of multistep methods on smooth nonuniform grids

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid $\{t_n\}_{n=0}^N$ can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., $t_n = Φ(τ_n)$, where $τ_n = n/N$ and the map $Φ$ is monotonically increasing with $Φ(0)=0$ and $Φ(1)=1$. The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines $Φ$, and a tolerance requirement which determines $N$. Given any strongly stable multistep method, there is an $N^*$ such that the method is zero stable for $N>N^*$, provided that $Φ\in C^2[0,1]$. Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy $h_n/h_{n-1} = 1 + \mathrm O(N^{-1})$ as $N\rightarrow\infty$. The results are exemplified for BDF-type methods.

math.NA↗

arXiv2011

Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $ε$, and the schemes under study preserve the symplectic structure in the case $ε=0$. In the case $0<ε\ll 1$ the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3--D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order.

math.NA↗