Heterogeneous multiscale methods for stiff ordinary differential equations

Abstract

Abstract. The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.

Mathematical test models for superparametrization in anisotropic turbulence

“…When the situation in Eq. 19 is satisfied for all values of u and the scale separation parameter, ε, is sufficiently small, the model is in a typical regime for application of HMM methods (14,19). In fact, one can write down a formal closure in the limit ε → 0 for any value of u that satisifies Eq.…”

“…HMM algorithms are a mathematical synthesis of earlier work (see refs. 15 and 16, and references therein) as well as an abstract formulation that leads to new multiscale algorithms for complex systems with widely disparate time scales (14,(17)(18)(19). However, as noted recently (8), there are significant differences in the regimes of nonlinear dynamics being modeled by superparametrization algorithms as compared with HMM.…”

“…However, as noted recently (8), there are significant differences in the regimes of nonlinear dynamics being modeled by superparametrization algorithms as compared with HMM. Reduced HMM time-steppers have been analyzed and applied for various physical systems with wide scale separation, ε = 10 −3 , 10 −4 , with ε the scale separation ratio between large and small scales, and rapid local statistical equilibration in time (17,19). However, the skill and success of superparametrization algorithms relies on intermittency in space and time due to complex evolving, strongly chaotic, and quiescent regions without statistical equilibration despite only modest values of scale separation, ε = 1 6 to 1 10 (8,12,13,20).…”

A Stroboscopic Numerical Method for Highly Oscillatory Problems

“…The process of obtaining the autonomous system (4) (or (8)) from the original system (1) is referred to in the averaging literature [13] as high-order stroboscopic averaging. As a rule, the amount of work required to find analytically the functions F j is formidable, even when the interest is limited to lowest values of j.…”

“…The new method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging [13] which constructs an averaged differential system dY /dt = F(Y ) whose solutions Y (approximately) interpolate the sought highly oscillatory solution y at times t = t 0 + 2πεn,(n integer). In the spirit of the heterogeneous multiscale methods (see [6,5,8,16,7,1], cf. [14,3]), we integrate numeri-cally the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the original system in small time windows.…”