Skipping The Jacobian

Abstract

We present a reduced order model (ROM) for a one-dimensional nonlinear gas dynamics problem: the isentropic piston. The main body of the PDE, the geometrical definition of the mesh nodes, and the boundary conditions are parametrized. The full order model is obtained with a Galerkin finite element discretization, under the Arbitrary Lagrangian Eulerian formulation (ALE). To stabilize the system, an artificial viscosity term is included. The nonlinear convective term is linearized with a second-order extrapolation. The reduced basis to express the solution is obtained with the POD technique. To overcome the explicit use of the Jacobian transformation, typical in the context of moving meshes, a system approximation technique is introduced. The (Matrix) Discrete Empirical Interpolation Method, (M)DEIM, allows us to work with a weak form defined in the physical domain (and hence the physical weak formulation) whilst maintaining an efficient assembly of the algebraic operators, despite their change with every time step. Two alternative methods are presented to collect and compress the snapshots for the discretized solution-dependent convective term. Each method leads to a different offline stage. All in all, our approach is purely algebraic and the reduced model makes no use of full order structures, thus achieving a perfect offline-online split. A concise description of the reduction procedure is provided. The reduced model is certified with a posteriori error estimations obtained via mode truncation.