Reduced Basis Method for Explicit Finite Volume Approximations of Nonlinear Conservation Laws

The numerical solution of parametrized partial differential equations (P2DEs ) can be a very time-consuming task if many parameter constellations have to be simulated by high-resolution schemes. Such scenarios may occur in parameter studies, optimization, control, inverse problems or statistical analysis of a given P2DE . Reduced Basis (RB) methods allow to produce fast reduced models that are good surrogates for the detailed numerical scheme and allow parameter variations. These methods have gained increasing attention in recent years for stationary elliptic and instationary parabolic problems. In the current presentation we present a RB method which is applicable to nonlinear conservation laws with explicit finite volume discretizations. We show that the resulting RB-method is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and runtimes. In all cases we obtain a runtime acceleration of at least one order of magnitude.