Accuracy analysis of acceleration schemes for stiff multiscale problems

Abstract

In the context of multiscale computations, techniques have recently been developed that enable microscopic simulators to perform macroscopic level tasks (equation-free multiscale computations). The main tool is the so-called coarse-grained time-stepper, which implements an approximation of the unavailable macroscopic time-stepper using only the microscopic simulator. Several schemes were developed to accelerate the coarse-grained time-stepper, exploiting the smoothness in time of the macroscopic dynamics. To date, mainly the stability of these methods was analyzed. In this paper, we focus on their accuracy properties, mainly in the context of parabolic problems. We study the global error of the different methods, compare with explicit stiff ODE solvers, and use the theoretical results to develop more accurate variants. Our theoretical results are confirmed by various numerical experiments.

A Micro-Macro Acceleration Method for the Monte Carlo Simulation of Stochastic Differential Equations

“…In particular, the time-scale separation will then manifest itself as a large gap in the spectrum, and the combination of microscopic simulation with extrapolation dumps the fast components, allowing for large ∆t. We refer to [26,71] for the study of the extrapolation procedure (3.7) in this context. Numerical implementation We present the Algorithm as it operates on the random variables, see also Remark 3.3 on how we understand the matching step in this framework.…”

“…In this manuscript we only consider first order extrapolation, which is reminiscent of forward Euler integration of the macroscopic state variables. This idea was first proposed in [27], see also [41,42,71].…”

Acceleration of lattice Boltzmann models through state extrapolation: a reaction–diffusion example

“…Using (9), it can be shown [34] that the (global) error E-which is affected by the local errors at each previous time step as well as by the propagation of these errors-is then roughly proportional to…”

“…For such time integrators, which also arise in the context of stiff ordinary differential equations [15], parabolic partial differential equations (PDEs) [21,38] or in multiscale computations [19], several numerical schemes were recently developed to accelerate the time integration process [10,11,34,35]. These schemes combine a number of time integration steps with an extrapolation method to compute a solution further ahead.…”

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

“…Higher order versions of (2.6), which require macroscopic states at additional time instances, can be constructed in several ways: using polynomial extrapolation [17]; implementing Adams-Bashforth or Runge-Kutta methods [35,36,42]; or trading accuracy for stability by designing a multistep state extrapolation method [46].…”