Learning second-order total variation diminishing flux limiters using differentiable solvers

This paper presents a data-driven framework for learning optimal second-order total variation diminishing (TVD) flux limiters via differentiable simulations. In our fully differentiable finite volume solvers, the limiter functions are replaced by neural networks. By representing the limiter as a pointwise convex linear combination of the Minmod and Superbee limiters, we enforce both second-order accuracy and TVD constraints at all stages of training. Our approach leverages gradient-based optimization through automatic differentiation, allowing a direct backpropagation of errors from numerical solutions to the limiter parameters. We demonstrate the effectiveness of this method on various hyperbolic conservation laws, including the linear advection equation, the Burgers' equation, and the one-dimensional Euler equations. Remarkably, a limiter trained solely on linear advection exhibits strong generalizability, surpassing the accuracy of most classical flux limiters across a range of problems with shocks and discontinuities. To further assess its practical applicability, the learned flux limiter is integrated into OpenFOAM and demonstrates excellent performance in three-dimensional flow problems. The learned flux limiters can be readily integrated into other existing computational fluid dynamics codes, and the proposed methodology also offers a flexible pathway to systematically develop and optimize flux limiters for complex flow problems. The code is available at https://github.com/BattModels/diff-phys-flux-limiter.