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This work presents a method for constructing online-efficient reduced modelsof large-scale systems governed by parametrized nonlinear scalar conservationlaws. The solution manifolds induced by transport-dominated problems such ashyperbolic conservation laws typically exhibit nonlinear structures, whichmeans that traditional model reduction methods based on linear approximationsare inefficient when applied to these problems. In contrast, the approachintroduced in this work derives reduced approximations that are nonlinear byexplicitly composing global transport dynamics with locally linearapproximations of the solution manifolds. A time-stepping scheme evolves thenonlinear reduced models by transporting local approximation spaces along thecharacteristic curves of the governing equations. The proposed computationalprocedure allows an offline/online decomposition and is online-efficient in thesense that the complexity of accurately time-stepping the nonlinear reducedmodel is independent of that of the full model. Numerical experiments withtransport through heterogeneous media and the Burgers' equation show orders ofmagnitude speedups of the proposed nonlinear reduced models based ontransported subspaces compared to traditional linear reduced models and fullmodels.

Manifold Approximations via Transported Subspaces: Model reduction for transport-dominated problems