A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids

A weighted essentially non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin finite element method on unstructured grids. The solution polynomials are reconstructed using a WENO scheme by taking advantage of handily available and yet valuable information, namely the derivatives, in the context of the discontinuous Galerkin method. The stencils used in the reconstruction involve only the van Neumann neighborhood and are compact and consistent with the DG method. The developed HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and time-accurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed HWENO limiter for the DG methods

A physics-based shock capturing method for unsteady laminar and turbulent flows

We present a shock capturing method for unsteady laminar and turbulent flows. The proposed approach relies on physical principles to increase selected transport coefficients and resolve unstable sharp features, such as shock waves and strong thermal and shear gradients, over the smallest distance allowed by the discretization. In particular, we devise various sensors to detect when the shear viscosity, bulk viscosity and thermal conductivity of the fluid do not suffice to stabilize the numerical solution. In such cases, the transport coefficients are increased as necessary to optimally resolve these features with the available resolution. The performance of the method is illustrated through numerical simulation of external and internal flows in transonic, supersonic, and hypersonic regimes.United States. Air Force. Office of Scientific Research (FA9550-16-1-0214)Pratt & Whitney Aircraft CompanyFundación Obra Social de La CaixaMassachusetts Institute of Technology. Office of the Dean for Graduate Education (Zakhartchenko Fellowship

Well-balanced rr-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations

In this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on $r$-adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solution is observed. Obtaining well-balanced numerical schemes for the shallow water equations on fixed meshes is nontrivial and has been a topic of much research. In [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887–1922] we introduced a well-balanced discontinuous Galerkin method using the theory of weak solutions for nonconservative products introduced in [G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548]. In this article we continue this approach and prove well-balancedness of a discontinuous Galerkin method for the shallow water equations on moving meshes. Numerical simulations are then performed to verify the $r$-adaptive method in combination with the space-time discontinuous Galerkin method against analytical solutions and showing its robustness on more complex problems

A short note on a 3D spectral analysis for turbulent flows on unstructured meshes

We propose two techniques for computing the energy spectra for 3D unstructured meshes that are consistent across different element types. These techniques can be particularly useful when assessing the dissipation characteristics and the suitability of several popular non-linear high-order methods for implicit large-eddy simulations (iLES). Numerical experiments demonstrate the performance of several element types for iLES of the Taylor-Green vortex, where a significantly different dissipation and dispersion mechanism for each element type is revealed. The energy spectra results are dependent on the technique selected for obtaining them, therefore an additional established technique from the literature is also included for comparison to further analyse their similarities and their differences. These techniques can be an integral tool for the tuning and calibration of non-linear high-order methods that can benefit both explicit and implicit large-eddy simulations (LES)

On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations

Finite element and finite difference discretizations for evolutionary convection-dif\-fusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods