(613h) Projective Integration With Adaptive Horizon and Error Control

A computationally efficient integration method for differential equations with multiple time-scales is presented. The approach utilizes two coupled integrators: an inner integrator to damp out fast dynamics and an outer integrator to project over the slow dynamics.¹ The inner-outer integration process is repeated until the desired integration is complete. The projection horizon of the outer integration needs to be both short enough to control error growth and long enough to appreciably speed up computational time. The projection horizon is normally found through experimentation as the trade-offs between projection error and speed up are problem specific. As projective integration is meant to speed up long simulations, this experimentation phase to identify the correct projection horizon can incur significant additional computational time.

In this paper we use adaptive control theory to develop a new approach to derive prediction models for the outer prediction model. Then approach we develop can be applied to linear and nonlinear (neural network type) models. We find that by fitting the simulation data generated during the inner integration to a linear or affine model, the projection horizon can be specified based on the user defined error tolerance and the stability of the linear or affine model. The outer integration can then be performed using the linear or affine model to project forward over the error controlled projection horizon.  

We discuss a few examples relevant to projective integration to show the utility of our error controlled projection horizon approach. The examples include a Brusselator problem, a stochastic polymer simulation, and an oil reservoir simulation which highlight different aspects of the algorithm such as handling problems of multiple time scales. For the Brusselator in particular, we are able to show two orders of magnitude of computational speed up over explicit Euler’s method. The polymer simulation shows how projective integration can be modified to solve problems with stochastic noise and the oil reservoir problem shows that the theory can be used to large scale simulation models, such as ECLIPSE for oil and gas filed simulation, that otherwise may take days or even weeks to solve. All cases rely on the fact that the underlying system models are dissipative so that the dynamics can be described accurately with a significantly reduced number of coordinates relative to the original large scale system.

1. Gear, C. W. and Kevrekidis, I. G. (2003). Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM Journal of Scientific Computing, 24(4) 1091 – 1106.