# An Implicit High-Order Preconditioned Flux Reconstruction Method for Low-Mach-Number Flow Simulation with Dynamic Meshes

## TL;DR

This paper introduces a fully implicit high-order preconditioned flux reconstruction method for low-Mach-number flow simulations with dynamic meshes, ensuring accuracy and efficiency in unsteady flow analysis.

## Contribution

It develops a novel implicit high-order preconditioned FR/CPR method combined with dual-time stepping and GCL enforcement for low-Mach flows with moving meshes.

## Key findings

- Demonstrates good convergence and accuracy in benchmark tests.
- Shows significant impact of weak compressibility on thrust generation.
- Validates the method's effectiveness for unsteady flow simulations.

## Abstract

A fully implicit high-order preconditioned flux reconstruction/correction procedure via reconstruction (FR/CPR) method is developed to solve the compressible Navier-Stokes equations at low Mach numbers. A dual-time stepping approach with the second-order backward differentiation formula (BDF2) is employed to ensure temporal accuracy for unsteady flow simulation. When dynamic meshes are used to handle moving/deforming domains, the geometric conservation law (GCL) is implicitly enforced to eliminate errors due to the resolution discrepancy between BDF2 and the spatial FR/CPR discretization. The large linear system resulted from the spatial and temporal discretizations is tackled with the restarted Generalized Minimal Residual (GMRES) solver in the PETSc (Portable, Extensible Toolkit for Scientific Computation) library. Through several benchmark steady and unsteady numerical tests, the preconditioned FR/CPR methods have demonstrated good convergence and accuracy for simulating flows at low Mach numbers. The new flow solver is then used to study the effects of Mach number on unsteady force generation over a plunging airfoil when operating in low-Mach-number flows. It is observed that weak compressibility has a significant impact on thrust generation but a negligible effect on lift generation of an oscillating airfoil.