Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method

This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate computation of these derivatives is important, for instance, in inverse problems and optimal control problems. This computation is usually based on an adjoint PDE system, and the question addressed in this paper is how the discretization of this adjoint system should relate to the dG discretization of the hyperbolic state equation. Adjoint-based derivatives can either be computed before or after discretization; these two options are often referred to as the optimize-then-discretize and discretize-then-optimize approaches. We discuss the relation between these two options for dG discretizations in space and Runge-Kutta time integration. Discretely exact discretizations for several hyperbolic optimization problems are derived, including the advection equation, Maxwell's equations and the coupled elastic-acoustic wave equation. We find that the discrete adjoint equation inherits a natural dG discretization from the discretization of the state equation and that the expressions for the discretely exact gradient often have to take into account contributions from element faces. For the coupled elastic-acoustic wave equation, the correctness and accuracy of our derivative expressions are illustrated by comparisons with finite difference gradients. The results show that a straightforward discretization of the continuous gradient differs from the discretely exact gradient, and thus is not consistent with the discretized objective. This inconsistency may cause difficulties in the convergence of gradient based algorithms for solving optimization problems

Algebraic construction and numerical behaviour of a new s-consistent difference scheme for the 2D Navier-Stokes equations

In this paper we consider a regular grid with equal spatial spacings and construct a new finite difference approximation (difference scheme) for the system of two-dimensional Navier-Stokes equations describing the unsteady motion of an incompressible viscous liquid of constant viscosity. In so doing, we use earlier constructed discretization of the system of three equations: the continuity equation and the proper Navier-Stokes equations. Then, we compute the canonical Gröbner basis form for the obtained discrete system. It gives one more difference equation which is equivalent to the pressure Poisson equation modulo difference ideal generated by the Navier-Stokes equations, and thereby comprises a new finite difference approximation (scheme). We show that the new scheme is strongly consistent. Besides, our computational experiments demonstrate much better numerical behaviour of the new scheme in comparison with the other strongly consistent schemes we constructed earlier and also with the scheme which is not strongly consistent