## Section: New Results

### Numerical methods for fluid flows

#### PARAOPT: A parareal algorithm for optimality systems

Member: J. Salomon,

*Coll.: Martin Gander, Felix Kwok*

The time parallel solution of optimality systems arising in PDE constraint optimization could be achieved by simply applying any time parallel algorithm, such as Parareal, to solve the forward and backward evolution problems arising in the optimization loop. We propose in [21] a different strategy by devising directly a new time parallel algorithm, which we call ParaOpt, for the coupled forward and backward non-linear partial differential equations. ParaOpt is inspired by the Parareal algorithm for evolution equations, and thus is automatically a two-level method. We provide a detailed convergence analysis for the case of linear parabolic PDE constraints. We illustrate the performance of ParaOpt with numerical experiments both for linear and nonlinear optimality systems.

#### Dynamical Behavior of a Nondiffusive Scheme for the Advection Equation

Member: N. Aguillon,

*Coll.: Pierre-Antoine Guihéneuf*

In [16], we study the long time behaviour of a dynamical system strongly linked to the anti-diffusive scheme of Després and Lagoutiere for the 1-dimensional transport equation. This scheme is overcompressive when the Courant–Friedrichs–Levy number is 1/2: when the initial data is nondecreasing, the approximate solution becomes a Heaviside function. In a special case, we also understand how plateaus are formed in the solution and their stability, a distinctive feature of the Després and Lagoutiere scheme.

#### Convergence of numerical schemes for a conservation equation with convection and degenerate diffusion

Member: C. Guichard,

*Coll.: Robert Eymard, Xavier Lhébrard*

In [20], the approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a $\theta $-scheme based on the centered gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of $\theta $ for stabilising the scheme.

#### Gradient-based optimization of a rotating algal biofilm process

Members: N. Aguillon, J. Sainte-Marie,

*Coll.: Pierre-Olivier Lamare, Jérôme Grenier, Hubert Bonnefond, Olivier Bernard*

Microalgae are microorganisms that have only very recently been used for bio-technological applications and more specifically for the production of bio-fuel. In the report [15] we focus on the shape optimization and optimal control of an innovative process where microalgae are fixed on a support. They are successively exposed to light and darkness. The resulting growth rate can be represented by a dynamic system describing the denaturation of key proteins due to excess light. A Partial Derivative Equation (PDE) model for the Rotary Algae Biofilm (RAB) is proposed. It represents the local growth of microalgae subjected to time-varying light. A gradient method based on the calculation of the model adjoint is proposed to identify the optimal (constant) folding of the process and the (time-varying) speed of the biofilm. Once this method is used in a realistic case, the optimization results in a configuration that significantly improves productivity compared to the case where the biofilm is fixed.