Model Reduction of Time-Dependent Hyperbolic Equations using Collocated Residual Minimisation and Shifted Snapshots
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We develop a non-linear approximation for solution manifolds of parametrised time-dependent hyperbolic PDEs. Our non-linear approximation space is a span of snapshots evaluated on a transformed spatial domain. We compute a solution in the non-linear approximation space using residual minimisation. We reduce the cost of residual minimisation by minimising and evaluating the residual on a set of collocation points. We decompose the collocation points computation into an offline and an online phase. The offline phase computes the collocation points for a set of training parameters by minimising a bound on the L2-error of the reduced-order model. Moreover, the online phase transports the collocation points computed offline. Our hyper-reduction is general in the sense that it does not assume a specific form of the spatial transform. As a particular instance of the non-linear approximation space, we consider a span of shifted snapshots. We consider shifts that are local in the time and parameter domain and propose an efficient algorithm to compute the same. Several benchmark examples involving single and multi-mode transport demonstrate the effectiveness and the limitations of our method.
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