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A time-adaptive algorithm for pressure dominated flows: a heuristic estimator

Ivan Prusak, Davide Torlo, Monica Nonino, Gianluigi Rozza

TL;DR

The paper addresses the challenge of efficiently solving pressure-dominated CFD and FSI problems where rapid pressure transients are poorly captured by traditional time-adaptive schemes. It introduces a time-adaptive algorithm based on a temporal error estimator that exploits the difference between BDF$2$ and BDF$3$ implicit solutions, augmented by a linear-implicit (LI) correction that uses a one-step Newton update to avoid solving the full BDF$3$ system. The step-size control combines a predictor with a tolerance-based update $\\Delta_t^*$ via $\\kappa^*=(\\varepsilon/est_{n+1})^{1/(q+1)}$ and a convex combination to form $\\Delta_t^{n+1}=\\alpha_0\\Delta_t^n+\\alpha_1\\Delta_t^*$, with $q=2$ and $(\\alpha_0,\\alpha_1)=(0.3,0.7)$. Numerical validation on a BFS backward-facing step at $\\Re=300$ and a 2D haemodynamics FSI benchmark shows accurate error control and meaningful computational savings, with the LI estimator delivering equivalent accuracy to the full implicit estimator at a substantially reduced cost.

Abstract

This work aims to introduce a heuristic timestep-adaptive algorithm for Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) problems where the flow is dominated by the pressure. In such scenarios, many time-adaptive algorithms based on the interplay of implicit and explicit time schemes fail to capture the fast transient dynamics of pressure fields. We present an algorithm that relies on a temporal error estimator using Backward Differentiation Formulae (BDF$k$) of order $k=2,3$. Specifically, we demonstrate that the implicit BDF$3$ solution can be well approximated by applying a single Newton-type nonlinear solver correction to the implicit BDF$2$ solution. The difference between these solutions determines our adaptive temporal error estimator. The effectiveness of our approach is confirmed by numerical experiments conducted on a backward-facing step flow CFD test case with Reynolds number $300$ and on a two-dimensional haemodynamics FSI benchmark.

A time-adaptive algorithm for pressure dominated flows: a heuristic estimator

TL;DR

The paper addresses the challenge of efficiently solving pressure-dominated CFD and FSI problems where rapid pressure transients are poorly captured by traditional time-adaptive schemes. It introduces a time-adaptive algorithm based on a temporal error estimator that exploits the difference between BDF and BDF implicit solutions, augmented by a linear-implicit (LI) correction that uses a one-step Newton update to avoid solving the full BDF system. The step-size control combines a predictor with a tolerance-based update via and a convex combination to form , with and . Numerical validation on a BFS backward-facing step at and a 2D haemodynamics FSI benchmark shows accurate error control and meaningful computational savings, with the LI estimator delivering equivalent accuracy to the full implicit estimator at a substantially reduced cost.

Abstract

This work aims to introduce a heuristic timestep-adaptive algorithm for Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) problems where the flow is dominated by the pressure. In such scenarios, many time-adaptive algorithms based on the interplay of implicit and explicit time schemes fail to capture the fast transient dynamics of pressure fields. We present an algorithm that relies on a temporal error estimator using Backward Differentiation Formulae (BDF) of order . Specifically, we demonstrate that the implicit BDF solution can be well approximated by applying a single Newton-type nonlinear solver correction to the implicit BDF solution. The difference between these solutions determines our adaptive temporal error estimator. The effectiveness of our approach is confirmed by numerical experiments conducted on a backward-facing step flow CFD test case with Reynolds number and on a two-dimensional haemodynamics FSI benchmark.
Paper Structure (5 sections, 15 equations, 3 figures, 4 tables)

This paper contains 5 sections, 15 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Time discretization and error estimator
  3. Backward Differentiation Formulae (BDF) with varying time-steps
  4. Time-adaptive algorithm and temporal error estimators
  5. Numerical simulations

Figures (3)

  • Figure 1: Domains of interest for the CFD (a) and FSI (b) test cases
  • Figure 2: FSI-H2 solution at time instance $t=0.01$ in the current fluid configuration
  • Figure 3: Adaptive time-steps distribution and relative errors w.r.t. constant-timestep solution (left) and the comparison of implicit and LI time estimators (right)