E, Weinan Analysis of the heterogeneous multiscale method for ordinary differential equations. (English) Zbl 1088.65552 Commun. Math. Sci. 1, No. 3, 423-436 (2003). From the summary: We analyze the heterogeneous multiscale method (HMM) for ordinary differential equations with multiple time scales. The analysis is an application of the general principle discussed by W. E and B. Engquist [The heterogeneous multiscale methods. Comm. Math. Sci. 1, No. 1, 86–134 (2003)], which states that HMM is stable if the macroscale solver is stable and the total error is a sum of the standard truncation error of the macroscale solver and the error in the \(F\)-estimator. Our analysis gives a fairly detailed understanding of how the error in the \(F\)-estimator depends on various components of HMM, such as the microscale solver and the filter, as well as the nature of the microscale problem. Cited in 40 Documents MSC: 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations Keywords:error estimates; heterogeneous multiscale method; multiple time scales; macroscale solver; \(F\)-estimator; microscale solver PDFBibTeX XMLCite \textit{W. E}, Commun. Math. Sci. 1, No. 3, 423--436 (2003; Zbl 1088.65552) Full Text: DOI