Heterogeneous Multiscale Methods Applied to Stiff Problems with Varying Scales.

摘要

This work explores fast numerical methods for solving rate equations that describe the population densities of chemical species or atomic states. The rate equations are very stiff nonlinear ordinary differential equations, with essentially one slow time scale and a large range of fast scales. We consider implicit multistep and one-step methods. They require the solution of a nonlinear system of equations in each time step with a Newton method. To reduce the cost of this, we use approximations or prefactorization of the Jacobian matrix. Different approximation strategies are explored. The importance of exact discrete conservation is highlighted, leading to an approach where the Jacobian is truncated to banded form and remaining off- diagonal elements are adjusted by a weight that depends on the elements in the full Jacobian. The prefactorization approach uses a QZ decomposition of the leading part of the Jacobian, and a separate treatment of a rank one part. Numerical experiments indicate that these methods give accurate results at a low computational cost.