Learning an Optimised Stable Taylor-Galerkin Convection Scheme Based on a Local Spectral Model for the Numerical Error Dynamics

We present a new spectral framework to design and optimise numerical methods for convection problems termed Local Transfer function Analysis (LTA), which improves the state-of-art model for error dynamics due to numerical discretisation. LTA converts the numerical discretisation into a network of transfer function blocks in the spectral space, where optimisation of the solution error becomes an impedance matching problem. In addition, a machine learning model based on the graph neural network is employed to learn the optimal coefficients of the transfer function leading to the matched impedance by examining both the local physical field and the local mesh topology. We demonstrate applications of this technique to design Taylor-Galerkin finite element numerical schemes for solving linear convection on irregular meshes in 1D. Using the LTA, we identify and constrain the neural network predictions to the numerically stable design space, thus improving the stability and generalisation of the optimal parameters that lead to a matched impedance.