A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

Abstract

A novel parameterized non-intrusive reduced order model (P -NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and e ffi cient approach for model reduction of parameterized partia l di ff eren- tial equations (P-PDEs). Over existing parameterized redu ced order models (P-ROM) (most of them are based on the reduced basis method), it is non -intrusive and inde- pendent on partial di ff erential equations and computational codes. During the tra ining process, the Smolyak sparse grid method is used to select a se t of parameters over a specific parameterized space ( Ω p ∈ R P ). For each selected parameter, the reduced ba- sis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function set s for any parameters over Ω p can be obtained using an interpolation method. The P-NIROM c an then be con- structed by using our recently developed technique [ 50 , 53 ] where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the redu ced space. The new P-NIROM technique has been applied to parameterized Navier-Stokes equations and implemented with an unstructured mesh finite e lement model. The ca- pability of this P-NIROM has been illustrated numerically b y two test cases: flow past a cylinder and lock exchange case. The prediction capabilit ies of the P-NIROM have been evaluated by varying the viscosity, initial and bounda ry conditions. The results show that this P-NIROM has captured the quasi-totality of th e details of the flow with CPU speedup of three orders of magnitude. An error analysis f or the P-NIROM has been carried out.