A New Class of Uniformly Accurate Numerical Schemes for Highly Oscillatory Evolution Equations

• Published in: Foundations of computational mathematics Vol. 20; no. 1; pp. 1 - 33
• Main Authors: Chartier, Philippe, Lemou, Mohammed, Méhats, Florian, Vilmart, Gilles
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2020; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Applications of Mathematics; Computer Science; Differential equations, Nonlinear; Economics; Evolution; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical research; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Stiffness; France; Standard averaging; Uniform accuracy; Highly oscillatory problems; Stroboscopic averaging; 65L20; 35K15; Pullback method; 74Q10; Micro–macro method; uniform accuracy AMS subject classification : 65L20; pullback method; standard averaging; uniform accuracy; micro-macro method; stroboscopic averaging; highly-oscillatory problems
• ISSN: 1615-3375; 1615-3383; 1615-3383
• DOI: 10.1007/s10208-019-09413-3

We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.