A note on statistical consistency of numerical integrators for multi-scale dynamics

TL;DR

This paper investigates how numerical integrators affect the statistical accuracy of simulating multi-scale systems, revealing that higher-order methods or corrected schemes are needed to reduce bias in the probability density functions of slow variables.

Contribution

It combines homogenization and backward error analysis to quantify statistical biases in numerical methods for multi-scale dynamics and proposes a corrected second-order scheme to mitigate bias.

Findings

Numerical integrators converge to the homogenized modified equations, not the exact pdf.

Bias is more pronounced in systems with slowly decorrelating chaotic fast dynamics.

A second-order Taylor method with a corrected vector field reduces statistical drift bias.

Abstract

A minimal requirement for simulating multi-scale systems is to reproduce the statistical behavior of the slow variables. In particular, a good numerical method should accurately aproximate the probability density function of the continuous-time slow variables. In this note we use results from homogenization and from backward error analysis to quantify how errors of time integrators affect the mean behavior of trajectories. We show that numerical simulations converge, not to the exact probability density function (pdf) of the homogenized multi-scale system, but rather to that of the homogenized modified equations following from backward error analysis. Using homogenization theory we find that the observed statistical bias is exacerbated for multi-scale systems driven by fast chaotic dynamics that decorrelate insufficiently rapidly. This suggests that to resolve the statistical behavior of trajectories in certain multi-scale systems solvers of sufficiently high order are necessary. Alternatively, backward error analysis suggests the form of an amended vector field that corrects the lowest order bias in Euler's method. The resulting scheme, a second order Taylor method, avoids any statistical drift bias. We corroborate our analysis with a numerical example.