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.