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We propose a novel model reduction approach for the approximation of nonlinear hyperbolic equations in the scalar and the system cases. The approachrelies on an offline computation of a dictionary of solutions together with anonline $L^1$-norm minimization of the residual. It is shown why this is anatural framework for hyperbolic problems and tested on nonlinear problems suchas Burgers' equation and the one-dimensional Euler equations involving shocksand discontinuities. Efficient algorithms are presented for the computation ofthe $L^1$-norm minimizer, both in the cases of linear and nonlinear residuals.Results indicate that the method has the potential of being accurate wheninvolving only very few modes, generating physically acceptable,oscillation-free, solutions.

Robust Model Reduction by $L^1$-norm Minimization and Approximation via Dictionaries: Application to Linear and Nonlinear Hyperbolic Problems