Hybridized Implicit-Explicit Flux Reconstruction Methods

TL;DR

The study addresses the computational challenge of geometry-induced stiffness in large-scale turbulent flows by marrying implicit-explicit time stepping with hybridized flux reconstruction. By placing the stiff, geometry-driven portion of the domain under a hybridized FR discretization and solving nonstiff regions explicitly, the method reduces the effective implicit problem size from $O(p^{d})$ to $O(p^{d-1})$ via static condensation. Across two- and three-dimensional benchmarks, including a high-Re multi-element airfoil, the hybridized IMEX approach achieves substantial speedups relative to standard IMEX and explicit FR, while maintaining accuracy comparable to fully implicit or FR solutions. The results suggest broad applicability of hybridized IMEX FR methods for industrial-scale Flow simulations and potential extensions to fluid-structure interaction and other stiffness-causing domain features.

Abstract

For turbulent problems of industrial scale, computational cost may become prohibitive due to the stability constraints associated with explicit time discretization of the underlying conservation laws. On the other hand, implicit methods allow for larger time-step sizes but require exorbitant computational resources. Implicit-explicit (IMEX) formulations combine both temporal approaches, using an explicit method in nonstiff portions of the domain and implicit in stiff portions. While these methods can be shown to be orders of magnitude faster than typical explicit discretizations, they are still limited by their implicit discretization in terms of cost. Hybridization reduces the scaling of these systems to an effective lower dimension, which allows the system to be solved at significant speedup factors compared to standard implicit methods. This work proposes an IMEX scheme that combines hybridized and standard flux reconstriction (FR) methods to tackle geometry-induced stiffness. By using the so-called transmission conditions, an overall conservative formulation can be obtained after combining both explicit FR and hybridized implicit FR methods. We verify and apply our approach to a series of numerical examples, including a multi-element airfoil at Reynolds number 1.7 million. Results demonstrate speedup factors of four against standard IMEX formulations and at least 15 against standard explicit formulations for the same problem.

Figures (20)

• Figure 1: Trace variable location in an HFR (left) and EFR (right) discretization considering a $p=2$ scheme on the skeleton of two neighbouring quadrilateral elements
• Figure 2: Reference domain partitioning for IMEX schemes
• Figure 3: Distribution of trace and solution points in this configuration for HFR and EFR IMEX methods for a $p=2$ discretization
• Figure 4: Distribution of explicit and implicit elements for the IMEX verification of linear advection
• Figure 5: Mesh for the laminar cylinder case at $\operatorname{Re}=150$ consisting of 3090 quadrilateral elements