A Semi-Lagrangian Computation of Front Speeds of G-equation in ABC and Kolmogorov Flows with Estimation via Ballistic Orbits

TL;DR

This paper develops a semi-Lagrangian numerical method to estimate front speeds in turbulent combustion models driven by ABC and Kolmogorov flows, using ballistic orbits and optimal control theory to improve accuracy.

Contribution

It introduces a novel approach combining ballistic orbit analysis with semi-Lagrangian schemes for precise front speed estimation in complex flows.

Findings

Front speed estimates approach analytical bounds at high flow intensities.

Semi-Lagrangian scheme with WENO interpolation effectively computes front speeds.

Ballistic orbits provide tight bounds for front speed in chaotic flows.

Abstract

The Arnold-Beltrami-Childress (ABC) flow and the Kolmogorov flow are three dimensional periodic divergence free velocity fields that exhibit chaotic streamlines. We are interested in front speed enhancement in G-equation of turbulent combustion by large intensity ABC and Kolmogorov flows. We give a quantitative construction of the ballistic orbits of ABC and Kolmogorov flows, namely those with maximal large time asymptotic speeds in a coordinate direction. Thanks to the optimal control theory of G-equation (a convex but non-coercive Hamilton-Jacobi equation), the ballistic orbits serve as admissible trajectories for front speed estimates. To study the tightness of the estimates, we compute the front speeds of G-equation based on a semi-Lagrangian (SL) scheme with Strang splitting and weighted essentially non-oscillatory (WENO) interpolation. Time step size is chosen so that the Courant number grows sublinearly with the flow intensity. Numerical results show that the front speed growth rate in terms of the flow intensity may approach the analytical bounds from the ballistic orbits.