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Preprint 11-2019

Nonlinear Galerkin Model Reduction for Systems with Multiple Transport Velocities

Source file is available as :   Portable Document Format (PDF)

Author(s) : Felix Black , Philipp Schulze , Benjamin Unger

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 11-2019

MSC 2000

37L65 Special approximation methods
65M15 Error bounds

Abstract :
We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a-posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

Keywords : transport dominated phenomena, model order reduction, error bound, nonlinear Galerkin, residual minimization

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