Abstract:
We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and provide preliminaries results toward the full weak convergence rate for non-self-adjoint linear operator. Key part of the proof does not rely on Malliavin calculus. Depending of the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.

Abstract:
the treatment of control problems governed by systems of conservation laws poses serious challenges for analysis and numerical simulations. this is due mainly to shock waves that occur in the solution of nonlinear systems of conservation laws. in this article, the problem of the control of euler flows in gas dynamics is considered. numerically, two semi-linear approximations of the euler equations are compared for the purpose of a gradient-based algorithm for optimization. one is the lattice-boltzmann method in one spatial dimension and five velocities (d1q5 model) and the other is the relaxation method. an adjoint method is used. good results are obtained even in the case where the solution contains discontinuities such as shock waves or contact discontinuities.