Multimesh iterative schemes and model reduction for linear and nonlinear advection–diffusion equations

The spatial discretization of conservation laws typically yields large-scale dynamical systems. As a result, simulating, analyzing or controlling such models requires a high computational cost. The goal of Model Order Reduction (MOR) consists in building a system of much lower complexity that approximates the dynamic of the problem accurately. The reduced model can be interpreted as a Petrov-Galerkin projection of the unsteady partial differential equations with nonlocal shape and test functions. For linear time-invariant dynamical systems, we analyze the efficiency of an iterative method that minimizes a measure of the approximation error. Using several discretizations on coarser meshes of the same problem, we propose a robust choice of initial condition for such a scheme. Furthermore, we describe extensions of this MOR scheme for linear time-varying and nonlinear models. Sparse large-scale linear systems of equations must be solved at each iteration of this algorithm. Unlike direct solvers, the computational complexity of iterative solvers can be linear with an appropriate preconditioner. For purely elliptic problems, the convergence of the Conjugate Gradient method with a multigrid preconditioner does not depend on the mesh size. The extensions to the Stokes and Navier--Stokes equations are investigated numerically.(FSA 3) -- UCL, 201