Adjoint DSMC for Nonlinear Spatially-Homogeneous Boltzmann Equation With a General Collision Model

TL;DR

This paper develops an adjoint method for the DSMC simulation of the spatially homogeneous Boltzmann equation with a broad class of collision models, extending previous work limited to Maxwell molecules.

Contribution

It introduces a new adjoint formulation that handles variable collision rates and general collision laws in DSMC, broadening the applicability of adjoint methods.

Findings

Derived a new adjoint formula for general collision models

Included a score function term in the adjoint equation

Captured collision parameter dependence with a new Jacobian matrix

Abstract

We derive an adjoint method for the Direct Simulation Monte Carlo (DSMC) method for the spatially homogeneous Boltzmann equation with a general collision law. This generalizes our previous results in [Caflisch, R., Silantyev, D. and Yang, Y., 2021. Journal of Computational Physics, 439, p.110404], which was restricted to the case of Maxwell molecules, for which the collision rate is constant. The main difficulty in generalizing the previous results is that a rejection sampling step is required in the DSMC algorithm in order to handle the variable collision rate. We find a new term corresponding to the so-called score function in the adjoint equation and a new adjoint Jacobian matrix capturing the dependence of the collision parameter on the velocities. The new formula works for a much more general class of collision models.