Controlling oscillations in high-order schemes using neural networks

High-order numerical solvers for conservation laws suffer from Gibbs phenomenon close to discontinuities, leading to spurious oscillations and a detrimental effect on the solution accuracy. A possible strategy to reduce their amplitude aims to add a suitable amount of artificial viscosity. Several models are available in the literature, which rely on the identification of a shock sensor, adding dissipation when the solution regularity is lost. The dependence on problem-specific parameters limits their performances. To solve this issue, in this thesis we propose a new technique based on artificial neural networks. In particular, we focus on the construction of multilayer perceptrons. Emphasis is given to the training phase, which is carried out using a robust dataset created using the classical models with optimal parameters. The online evaluation is then integrated in the numerical solver of the partial differential equation. Even though most of the effort is put in one-dimensional problems, an extension to a two-dimensional scenario is provided. Several numerical results are presented to demonstrate the capabilities of the network-based technique. Initially, we focus on one-dimensional scalar problems, where Burgers equation represents our first benchmark test. Then, we move to less simple cases, characterized by non-convex flux functions or involving multiple equations (compressible Euler equations). The same strategy is followed for multi-dimensional problems. In most of the cases, the proposed model is able to guarantee high accuracy in presence of smooth solutions and to capture discontinuities (shock and contact waves). In general, the results are comparable to (or better than) the classical models with properly tuned parameters. A final performance analysis is carried out.