Continuous finite element methods for under resolved turbulence in compressible flow

Lead Research Organisation: University College London
Department Name: Mathematics


Here we will give a summary of the main research questions of the present project and how we intend to address them.
1. Three dimensional compressible flow problems combine the instabilities known from incompressible flows such as turbulence with the effects of acoustics in the subsonic regime and nonlinear waves such as shocks, contact discontinuities and rarefactions in the supersonic regime. How can a method be designed that handles all these phenomena in a unified way while remaining computationally efficient?
2. The analysis of numerical methods for compressible flow is typically restricted to asymptotic estimates for scalar problems. In part due to the lack of theoretical understanding of the continuous equations, but even for linear model problems the non-linear shock capturing type schemes necessary to suppress local oscillations, remain poorly understood. Can a more complete numerical analysis be carried out if additional assumptions are made on the exact solution, for instance that the solution can be decomposed in a finite number of smooth parts, separated by discontinuities, where the support of these discontinuities has some favourable properties?
3. In practice computational efficiency is of essence for large-scale computations. Aspects of numerical stability and the possibility to use explicit time-stepping have made the discontinuous Galerkin method popular for compressible flow computations. However, in two space dimensions piecewise affine discontinuous approximation has six times as many degrees of freedom (dofs) as standard continuous FEM. This number increases to 20 times in three dimensions. The discontinuous Galerkin method also relies on expensive Riemann solvers that may not always be robust, see [Abg17a].
Therefore, we ask if a continuous finite element method can be designed that incorporates the advantages of the discontinuous Galerkin using substantially fewer dofs [Gue16a,Abg17b]?

The main aim of the present project is to address these questions drawing on our recent results on finite element methods (FEM) for approximating of turbulent flows in the approximation of the incompressible Navier-Stokes' equations [Mou22], local estimates for stabilized FEM for scalar linear transport problems [Bu22a] and global estimates for scalar linear transport problems discretized with nonlinear stabilization [Bu22b]. The cornerstones of the present proposal are:
1. development of invariant preserving shock capturing methods for nonlinear waves;
2. development of linear stabilisation methods for the control of secondary oscillations;
3. development and numerical analysis of explicit and implicit-explicit time discretisation schemes;
4. load balanced domain decomposition methods for the mass matrix, for explicit time-stepping;
5. High performance three dimensional computations of compressible turbulent flows.

[Abg17a] Abgrall, R. Some failures of Riemann solvers. Handbook of numerical methods for hyperbolic problems, 18 2017.
[Abg17b] Abgrall, R. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. J. Sci. Comput. (2017).
[Bu22a] Burman, E. Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps. Vietnam J. Math. (2022).
[Bu22b] Burman, E. Some observations on the interaction between linear and nonlinear stabilization for continuous finite element methods applied to hyperbolic conservation laws. To appear in SIAM J. Sci. Comput.
[Gue16a] Guermond, J.-L.; Popov, B. Error estimates of a first-order Lagrange finite element technique for nonlinear scalar conservation equations. SIAM J. Numer. Anal. (2016).
\item[Mou22] Moura, R. C.; Cassinelli, A.; da Silva, A.; Burman, E.; Sherwin, S. Gradient jump penalty stabilisation of spectral/hp element discretisation for under-resolved turbulence simulations. CMAME (2022)


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