A Space-Time Certified Reduced Basis Method for Burgers’ Equation

We present a space-time certified reduced basis method for Burgers’ equation over the spatial interval (0, 1) and the temporal interval (0, T ] parametrized with respect to the Peclet number. We first introduce a Petrov-Galerkin space-time finite element discretization, which enjoys a favorable inf-sup constant that decreases slowly with Peclet number and final time T . We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi-Rappaz-Raviart a posteriori error bounds. We detail computational procedures that permit offline-online decomposition for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf-sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T , unlike the exponentially growing estimate of the classical formulation for high Peclet number cases.