A parallel-in-time solver for nonlinear degenerate time-periodic parabolic problems
Herbert Egger, Andreas Schafelner
TL;DR
This work addresses the numerical solution of degenerate nonlinear time-periodic parabolic problems arising in engineering by proposing a fixed-point, time-discretized solver where each step solves a linear time-periodic, time-invariant system that can be addressed with parallel-in-time techniques. A key contribution is proving global convergence with a contraction factor that is independent of discretization parameters, achieved by selecting a linear operator $\widehat{\mathcal{K}}$ (e.g., $\widehat{\mathcal{K}} = \frac{1}{\omega}\mathcal{R}^{-1}$) so that the inner contraction bound $q(\omega)$ satisfies $q(\omega)^2 = 1 - 2\omega\gamma + \omega^2 L^2$ for $0 < \omega < L^2/\gamma$, and using either a discrete Fourier transform or multigrid-in-time for the linear solves. The method is initialized with decoupled nonlinear static problems and is demonstrated on a 2D power-transformer model, showing robust convergence and competitive performance compared to Newton-based and simple time-stepping approaches. The results indicate that the proposed parallel-in-time solver provides a provably convergent, scalable framework for nonlinear time-periodic PDEs with applications in electrical engineering and related fields. Overall, the approach enables efficient, discretization-parameter-independent convergence and effective parallelization for industrial-scale time-periodic nonlinear problems.
Abstract
A class of abstract nonlinear time-periodic evolution problems is considered which arise in electrical engineering and other scientific disciplines. An efficient solver is proposed for the systems arising after discretization in time based on a fixed-point iteration. Every step of this iteration amounts to the solution of a discretized time-periodic and time-invariant problem for which efficient parallel-in-time methods are available. Global convergence with contraction factors independent of the discretization parameters is established. Together with an appropriate initialization step, a highly efficient and reliable solver is obtained. The applicability and performance of the proposed method is illustrated by simulations of a power transformer. Further comparison is made with other solution strategies proposed in the literature.