Russel Caflisch

Hayden Schaeffer, Stanley Osher, Russel Caflisch et al.

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high frequency source terms.

Russel Caflisch, Yunan Yang

This manuscript derives adjoint equations for the numerical solution of the spatially inhomogeneous Boltzmann equation using Direct Simulation Monte Carlo (DSMC). The formulation accounts for spatial transport and a range of boundary conditions, including periodic boundaries, specular reflection, thermal reflection, and prescribed inflow. Numerical experiments are presented to validate the resulting adjoint system. These adjoint formulations are intended for use in gradient-based optimization, sensitivity analysis, and design problems involving rarefied gas dynamics.