Towards high fidelity numerical simulations: machine learning and numerical analysis

Many engineering and scientific problems can be described by equations of fluid dynamics, namely, systems of time dependent nonlinear hyperbolic PDEs. The mathematical description of these processes, as well as the numerical discretisation of the resulting PDEs, will depend on the level of detail required to study them.

In this talk I want to present two approaches towards higher fidelity numerical simulations: 1) we developed fast and accurate data-driven surrogate models to emulate planetary collisions, with the objective to combine machine learning (ML) and traditional computational fluid dynamics solvers to account for multi-scales physics. 2) We developed a novel arbitrarily high-order (in space and time) numerical scheme to solve the induction equation that guarantees a divergence free representation of the magnetic field. This method is then extended to include shock-capturing capabilities through an a posteriori limiting strategy, allowing us to accurately evolve nonlinear systems of conservation laws, with the ultimate goal of solving the magneto-hydrodynamics equations with arbitrary high-order accuracy and a divergence free representation of the magnetic field.