Controlling oscillations in spectral schemes using Artificial Neural Networks

Global Fourier spectral methods are excellent tools to solve conserva- tion laws. They enable fast convergence rates and highly accurate solutions. However, being high-order methods, they suffer from the Gibbs phenomenon, which leads to spurious numerical oscillations in the vicinity of discontinuities. This can have a detrimental effect on the solution quality and lead to unphysical results. While local approxima- tion techniques allow for local limiting or reconstruction, there are no such possibilities for global methods. This thesis proposes a neural net- work based method that adds artificial viscosity around discontinuities of the solution to the conservation law. This enables the transforma- tion of discontinuities into steep but continuous jumps. Test cases in one and two spatial dimensions as well as systems of conservation laws (Euler equations) are solved. Furthermore, the method is generalized to other global basis approaches on non-uniform grids and reduced ba- sis methods. The proposed method delivers satisfactory results in all test cases. On the one hand, it is able to detect and handle discontinu- ities. On the other hand, it stays highly accurate for smooth data.