Discontinuous Galerkin Trefftz Methods for Model Reduction of Wave Phenomena

ABSTRACT The space–time discontinuous Galerkin (dG)‐Trefftz is known to be a highly efficient numerical scheme for solving linear hyperbolic problems. We investigate to what extent such a dG‐Trefftz method can be used as a basis for a model reduction method for a traveling wave problem using the wave speed as a parameter. Such problems are known to be tough for linear model reduction techniques, as the error decay is slow with increasing size of the reduced model (by the Kolmogorov ‐width). The presented dG‐Trefftz method yields a nonlinear model reduction technique as the reduced trial space is parameter‐dependent. We present results of numerical experiments that show a convergence rate that is better than the known worst‐case rate for linear schemes, but which is still polynomial. We compare the dG‐Trefftz method with a nonlinear model technique based upon trained autoencoders using neural networks. It turns out that these methods are not able to outperform the dG‐Trefftz scheme.