Approximate linear response for slow variables of dynamics with explicit time scale separation

Many real-world numerical models are notorious for the time scale separation of different subsets of variables and the inclusion of random processes. The existing algorithms of linear response to external forcing are vulnerable to the time scale separation due to increased response errors at fast scales. Here we develop the approximate linear response algorithm for slow variables in a two-scale dynamical system with explicit separation of slow and fast variables, which has improved numerical stability and reduced computational expense.