High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows (1508.04245v2)

Abstract: In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the disctinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment. Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem. For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: H(div)-conforming finite elements to garantuee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems.

• The paper presents a high-order HDG formulation that produces exactly divergence-free velocity solutions for unsteady Navier-Stokes flows.
• It employs operator-splitting to separately treat stiff linear and nonlinear components, optimizing both accuracy and efficiency.
• The introduction of projected jumps reduces the polynomial degree of facet unknowns, yielding significant computational gains.

High Order Divergence-Free HDG Methods for Unsteady Incompressible Flows

Christoph Lehrenfeld and Joachim Schöberl present a robust and efficient numerical method for solving the unsteady incompressible Navier-Stokes equations. The paper introduces a high-order Hybrid Discontinuous Galerkin (HDG) formulation specifically tailored for complex geometries and meshes featuring curved elements. The method enhances computational efficiency by distinguishing between stiff linear components and nonlinear components, thereby optimizing their temporal and spatial treatment using operator-splitting strategies.

Methodology

The proposed methodology employs:

1. Hybrid Discontinuous Galerkin Methods: The HDG approach is utilized for the spatial discretization, ensuring exactly divergence-free solutions. It hinges on $H(div)$-conforming finite elements, resulting in velocity solutions that are energetically stable and naturally satisfy the incompressibility constraint. The HDG framework introduces facet unknowns that aid in reducing the number of globally coupled unknowns, making the method computationally more viable than traditional DG approaches.
2. Operator-Splitting Time Integration Schemes: This method splits the Navier-Stokes problem into simpler sub-problems: a hyperbolic transport problem and an unsteady Stokes problem. An upwind DG method is applied to the hyperbolic transport component, allowing for explicit time integration, while the unsteady Stokes component is handled using implicit methods that benefit from the HDG formulation.
3. Projected Jumps: The paper presents a novel enhancement to HDG methods through the concept of projected jumps, which reduces the polynomial degree of facet unknowns without compromising solution accuracy. This refinement leads to efficiency gains, particularly in the context of high-order polynomial approximations.

Results and Discussion

The authors substantiate the effectiveness of the method through several benchmark numerical examples, demonstrating optimal convergence rates in various $L^2$ norms and robust performance in both two-dimensional and three-dimensional flow scenarios. They note the considerable reduction in computational complexity when using their HDG methodology with projected jumps, compared to traditional DG methods. The reduced stabilization technique effectively decreases the number of coupling points and improves computational efficiency.

Implications and Future Work

This hybridized approach to solving the incompressible Navier-Stokes equations contributes significantly to the computational fluid dynamics (CFD) sphere, offering a balance between accuracy and efficiency. It is particularly beneficial in handling high Reynolds number flows where convection dominates. The method's compatibility with sophisticated geometric configurations opens avenues for its application in real-world engineering problems, including aerodynamics and hydrodynamics simulations.

Further research could explore the extension of these HDG methods to other types of partial differential equations beyond fluid dynamics. Additionally, developing preconditioning techniques for the linear systems arising from these discretizations will be crucial in facilitating their application to large-scale industrial problems.

In conclusion, the paper firmly positions high-order divergence-free HDG methods with operator-splitting time integrations as a potent tool for unsteady flow simulations, providing a framework that enhances computational efficiency without sacrificing solution fidelity.