Improving the stability and accuracy of discontinuous Galerkin schemes using neural networks
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The discontinuous Galerkin method allows very accurate numerical approximations of regular solutions to hyperbolic equations. However, it is more difficult to approximate discontinuous solutions or perturbations around stationary solutions (for equations with source terms). In the first case, Gibbs oscillations generated at discontinuities can destabilise the scheme, while in the second case, the error produced on the stationary solution makes it difficult to study perturbative dynamics. In this presentation, we will see how neural networks can be used to construct artificial viscosities that stabilise numerical schemes and how they can be used to construct Discontinuous Galerkin bases adapted to the stationary solutions of the problem.
