Abstract
We present an adaptive local time-stepping (ALTS) scheme for a block-structured multiresolution scheme of hyperbolic conservation laws for fluid flow. The stability of standard local time-stepping (LTS) schemes with level-dependent time-step sizes is improved by local time-step size adaptation when progressing through the underlying multi-stage time integration scheme. The novelty of the approach is that it merges flux computation and time integration of the state vector with projection and prediction operations of the multiresolution scheme [15]. This enables consistent time integration of subdomains with different refinement levels without the need for intermediate time synchronization which can be prohibitively expensive in parallel computations. Consequently, coarser subdomains are advanced in time only once finer subdomains have advanced to the same time instant. Full spatial resolution adaptivity for integrated regions after each substep is maintained.The new scheme exhibits significantly improved numerical stability as compared to previous LTS schemes due to the local time-step size adaptation at each substep. The computational overhead of the incurred additional operations is small. In applications, the ALTS scheme demonstrates the same computational efficiency as standard LTS schemes.The new scheme can be applied to any explicit single-step time-integration scheme and is independent of the employed spatial discretization scheme. The improved stability is demonstrated with several one- and two-dimensional examples of flows with one and two phases, applying second- and third-order Runge-Kutta time integration schemes.
Highlights
Computational simulations of complex flows remain a challenge even today
We propose a fully adaptive local time-stepping (ALTS) algorithm for multiresolution simulations to solve the aforementioned limitations
Operations of a full Runge-Kutta cycle are split into three main categories: the update of the time-step size, additional operations that were introduced with the ALTS algorithm (ALTS related), and operations that are required for both LTS and ALTS schemes
Summary
Computational simulations of complex flows remain a challenge even today. The large range of temporal and spatial scales in combustive or turbulent flows or flow cavitation challenges the performance of even the largest parallel computers [26]. Müller and Stiriba [24] introduced transition zones between coarser and finer cells This enables updating the time-step size after each full cycle on the finest levels and propagating this change upwards to coarser grid regions. Their scheme has been successfully employed for multi-fluid simulations [5]. The time-step size is adapted after each full cycle on the finest level, recovering the stability properties of the CTS approach This is not limited to single-stage time-integration schemes, and applies to multi-stage time integration schemes, e.g., the twostage second order Runge-Kutta (RK) Total Variation Diminishing (TVD) scheme [11,14].
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