High-order temporal accuracy for 3D finite-element ALE flow simulations

• We consider FE solution of the 3D Navier–Stokes equations on deforming domains.
• The proposed ALE formulation allows the use of any time integrator.
• We present thorough stepsize refinement studies to assess orders of convergence.
• Results are reported for the variable stepsize implicit BDF and Radau IIA IRK.

Context: Many engineering problems require to solve PDE on deforming domains to account for the temporal evolution of the domain boundaries. Fluid–Structure Interaction problems and free-surface flows solved by a front-tracking approach are two such examples.Objective: In this paper, we address the numerical solution of the three-dimensional Navier–Stokes equations on deforming domains using the finite-element (FE) method and an Arbitrary Lagrangian Eulerian (ALE) formulation.Method: The proposed formulation incorporates the ALE mapping into the finite-element method in a natural and straightforward manner. It allows the use of any time integrator that can be expressed as a finite-difference in time and maintains the integrators fixed grid convergence rate on ALE deforming grids. Hence, popular time integrators (implicit backward Euler, Gear, BDF, Runge–Kutta, etc.) can be applied directly to moving grid simulations without necessitating any modification or adjustment to the code.Results: Using a manufactured solution, we present thorough time stepsize refinement studies. Results are reported for two families of time-stepping procedures: the implicit Backward Differentiation Formulas (multi-step methods of order 1–5) and the Implicit (Radau IIA) Runge–Kutta methods (one-step methods of order 1, 3 and 5).Conclusion: The proposed FE/ALE formulation preserves the fixed-grid orders of accuracy of time-stepping procedures on 3D moving grids. Hence, three-dimensional FE/ALE simulations can be performed with highly accurate time integrators.