Topological regularization on numerical simulations of the advection equation

Abstract

Numerical simulation of the one-dimensional advection equations is known to be difficult with discontinuous initial states. The initial shape undergoes distortion due to high-frequency fluctuations, exemplifying a prevalent challenge in discretizing dynamics: the violation of conservation laws. In addressing this issue, our study introduces a topological regularization method for numerical simulations, leveraging persistent homology. The effectiveness of our approach is demonstrated through numerical simulations of the one-dimensional advection equation with a rectangular initial state. The results highlight the potential of our method to improve the accuracy and fidelity of simulations, especially in scenarios where maintaining topological conservation is critical.