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We investigate the possibility of reducing the computational burden of LES models by employing local polynomial degree adaptivity in the framework of a high order DG method. A novel degree adaptation technique especially featured to be effective for LES applications is proposed and its effectiveness is compared to that of other criteria already employed in the literature. The resulting locally adaptive approach allows to achieve significant reductions in computational cost of representative LES computations., 39 pages, 10 figures, 5 tables

The Hybrid RANS/LES method proposed is based on the hybrid filter approach introduced by Germano in 2004. In this work, instead of using two explicit models respectively for LES and RANS, we use only a model for LES and we reconstruct the RANS field exploiting the properties of hybrid filter. The reconstruction is obtained from the resolved velocity and the LES subgrid stress tensor. The model has been implemented using a variational approach in a DG-FEM framework and tested for the turbulent channel flow test case. Different configurations for hybrid terms have been compared with DNS data and with pure LES results.

A semi-implicit and semi-Lagrangian discontinuous Galerkin method for the shallow water equations is proposed, for applications to geophysical scale flows. A non conservative formulation of the advection equation is employed, in order to achieve a more treatable form of the linear system to be solved at each time step. The method is equipped with a simple p-adaptivity criterion, that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution. Numerical results show that the method captures well the main features of gravity and inertial gravity waves, as well as reproducing correct solutions in nonlinear test cases with analytic solutions. The accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high Courant numbers and with automatic choice of the local approximation degree.

This work provides the theoretical formulation, implementation directives and validation case for the extension of the nonlinear model predictive control from the time domain to the spatio-temporal domain. Effectiveness and online feasibility are demonstrated for industrial processes of distributed nature. The case of methanol synthesis (control of hot-spot temperature and axial position) is selected.

The Discontinuous Galerkin Coastal Ocean Model (DGCOM) is a two-dimensional shallow water model for simulating coastal ocean processes. The article of record may be found at http://www.doi.org/10.1002/fld.2118 We extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI time-integrators using the backward difference formulas from orders one to six. The reason for changing the time-integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high-order DG methods. Changing the time- integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element-type area integrals, but also the finite volume-type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time-integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high-order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time-explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high-order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high-order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high-order (HO) time-integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low-order time discretizations.

A new, conservative semi-Lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form. The approach combines the accuracy and conservation properties of the Discontinuous Galerkin (DG) method with the computational efficiency and robustness of Semi-Lagrangian (SL) techniques. Unconditional stability in the von Neumann sense is proved for the proposed discretization in the one-dimensional case. A monotonization technique is then introduced, based on the Flux Corrected Transport approach. This yields a multi-dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard DG methods. The accuracy and stability of the method are further demonstrated by two-dimensional tracer advection tests in the case of incompressible flows. The comparison with results obtained by standard SL and DG methods highlights several advantages of the new technique.