In this work, we adapt the {\em micro-macro} methodology to stochastic differential equations for the purpose of numerically solving oscillatory evolution equations. The models we consider are addressed in a wide spectrum of regimes where oscillations may be slow or fast. We show that through an ad-hoc transformation (the micro-macro decomposition), it is possible to retain the usual orders of convergence of Euler-Maruyama method, that is to say, uniform weak order one and uniform strong order one half. We also show that the same orders of uniform accuracy can be achieved by a simple integral scheme. The advantage of the micro-macro scheme is that, in contrast to the integral scheme, it can be generalized to higher order methods.
A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations
INRIA Rennes, IRMAR and ENS Rennes（法国国家信息与自动化研究所雷恩分部、IRMAR和雷恩高等师范学院）；CNRS, IRMAR and ENS Rennes（法国国家科学研究中心、IRMAR和雷恩高等师范学院）；Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625（雷恩大学、法国国家信息与自动化研究所、法国国家科学研究中心、IRMAR联合实验室）；Université de Genève（日内瓦大学）
Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schrödinger equations
INRIA-Rennes Bretagne Atlantique（法国国家信息与自动化研究所雷恩布列塔尼大西洋中心）；IPSO Project（IPSO项目组）；CNRS and IRMAR, Université de Rennes 1（法国国家科学研究中心与雷恩第一大学IRMAR）；IRMAR, Université de Rennes 1（雷恩第一大学IRMAR）
Highly-oscillatory problems with time-dependent vanishing frequency
Univ Rennes, INRIA, IRMAR - UMR 6625（雷恩大学，INRIA，IRMAR）；Univ Rennes, CNRS, IRMAR - UMR 6625（雷恩大学，CNRS，IRMAR）；Université de Genève, Section de mathématiques（日内瓦大学，数学系）
In this paper we present a novel class of asymptotic consistent exponential-type integrators for Klein-Gordon-Schrödinger systems that capture all regimes from the slowly varying classical regime up to the highly oscillatory non-relativistic limit regime. We achieve convergence of order one and two that is uniform in $c$ without any time step size restrictions. In particular, we establish an explicit relation between gain in negative powers of the potentially large parameter $c$ in the error constant and loss in derivative.
The value of a highly oscillatory integral is typically determined asymptotically by the behaviour of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary points or saddle points -- roots of the derivative of the phase of the integrand -- where the integrand is locally non-oscillatory. Modern methods for highly oscillatory quadrature exhibit numerical issues when two such saddle points coalesce. On the other hand, integrals with coalescing saddle points are a classical topic in asymptotic analysis, where they g…
An exponential integrator for a highly oscillatory Vlasov equation
Université Européenne de Bretagne, LMBA (UMR CNRS 6205), Université de Bretagne-Sud（欧洲布列塔尼大学，LMBA（CNRS联合研究单位6205），南布列塔尼大学）；Inria Nancy-Grand Est, CALVI Project & IRMA (UMR CNRS 7501), Université de Strasbourg（法国国家信息与自动化研究所南斯特拉斯堡分部，CALVI项目，IRMA（CNRS联合研究单位7501），斯特拉斯堡大学）；Max Planck Institute for Plasma Physics, EURATOM Association（马克斯·普朗克等离子体物理研究所，EURATOM协会）
We propose a novel class of uniformly accurate integrators for the Klein--Gordon equation which capture classical $c=1$ as well as highly-oscillatory non-relativistic regimes $c\gg1$ and, at the same time, allow for low regularity approximations. In particular, the schemes converge with order $τ$ and $τ^2$, respectively, under lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods, require. The new schemes in addition preserve the nonlinear Schrödinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way t…
Oscillation Preserving Galerkin Methods for Fredholm Integral Equations of the Second Kind with Oscillatory Kernels
College of Science, National University of Defense Technology（国防科技大学理学院）；Department of Mathematics, Syracuse University（雪城大学数学系）；Guangdong Province Key Lab of Computational Science, School of Mathematics and Computational Science, Sun Yat-Sen University（中山大学数学与计算科学学院）
Uniformly accurate exponential-type integrators for klein-gordon equations with asymptotic convergence to classical splitting schemes in the nonlinear schroedinger limit
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；INRIA & ENS Rennes（法国国家信息与自动化研究所与雷恩高等师范学院）
High-order uniformly accurate fully discrete schemes with long time near-conservations for two nonlinear coupled dispersive models in the nonrelativistic regime
Xi’an Jiaotong University（西安交通大学）；Hebei Normal University（河北师范大学）
In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequenci…
