Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Abstract

Abstract. Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of high-order methods amongst a non-specialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and finally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruction approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat towards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

Nodal Points and the Nonlinear Stability of High-Order Methods for Unsteady Flow Problems on Tetrahedral Meshes

“…In general, upon expanding the second term on the left hand side of equation (1), one obtains second derivatives of the solution u. These second derivatives are difficult to discretize, and therefore equation (1) is usually rewritten as a system of two equations each of which contain first derivatives.…”

“…High-order methods produce less numerical dissipation than their low-order counterparts (for which the order is ≤ 2), and as a result, are well-suited for simulating vortex-dominated flows 1 . In particular, high-order methods have been successfully employed to simulate flows over flapping wings 2,3 , rotorcraft 4 , turbine blades 5 , and high-lift devices 6 .…”

Efficient finite element assembly of high order whitney forms

“…There is a growing consensus in the finite element community that state of the art low order finite element technology requires, and will continue to require, too extensive computational resources to provide the necessary resolution for complex simulations, even at the rate of computational power increase [1]. The requirement for precise resolution naturally leads us to consider methods with a higher order of grid convergence than the classical second order provided by most industrial grade codes.…”

An Extension of Energy Stable Flux Reconstruction to Unsteady, Non-linear, Viscous Problems on Mixed Grids

“…Further research efforts are required to facilitate their wide-spread adoption. 15 Currently, the majority of research efforts are focused on Discontinuous Galerkin (DG) methods. The DG approach was first used by Reed and Hill 16 in 1973 for the solution of the hyperbolic neutron transport equation.…”