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second-order schemes for solving decoupled forward backward stochastic differential equations

Science China Mathematics 2014年第4期57卷 665-686页

作者： ZHAO WeiDong LI Yang FU Yu School of Mathematics Shandong University College of Science University of Shanghai for Science and Technology

In this paper,by using trapezoidal rule and the integration-by-parts formula of Malliavin calculus,we propose three new numerical schemes for solving decoupled forward-backward stochastic differential *** theoretically prove that the schemes have second-order convergence *** demonstrate the effectiveness and the second-order convergence rate,numerical tests are given.

关键词： forward backward stochastic differential equations second-order scheme error estimate trape-zoidal rule Malliavin calculus

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SEMI-IMPLICIT SPECTRAL DEFERRED CORRECTION METHODS BASED ON second-order TIME INTEGRATION schemeS FOR NONLINEAR PDES

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Journal of Computational Mathematics 2024年第1期42卷 111-133页

作者： Ruihan Guo Yan Xu School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China

In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional *** this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional *** spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and *** experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

关键词： Spectral deferred correction method Nonlinear PDEs Local discontinuous Galerkin method second-order scheme

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A Parallel second order Cartesian Method for Elliptic Interface Problems

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Communications in Computational Physics 2012年第10期12卷 1562-1587页

作者： Marco Cisternino Lisl Weynans Dipartimento di Ingegneria Aeronautica e Spaziale Politecnico di Torino C.so Duca degli Abruzzi 2410129 TorinoItaly Univ.Bordeaux IMBUMR 5251F-33400 TalenceFrance.CNRSIMBUMR 5251F-33400 TalenceFrance.INRIAF-33400 TalenceFrance

We present a parallel Cartesianmethod to solve elliptic problems with complex immersed *** method is based on a finite-difference scheme and is second-order accurate in the whole *** originality of the method lies in the use of additional unknowns located on the interface,allowing to express straightforwardly the interface transmission *** describe the method and the details of its parallelization performed with the PETSc *** we present numerical validations in two dimensions,assorted with comparisons to other related methods,and a numerical study of the parallelized method.

关键词： Elliptic interface problem Cartesian method second-order scheme interface unknowns

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Spatial second-order positive and asymptotic preserving unified gas kinetic schemes for radiative transfer equations

Spatial second-order positive and asymptotic preserving unif...

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第十二届全国流体力学学术会议

作者： Xiaojing Xu Song Jiang Wenjun Sun School of Mathematics and Physics Southwest University of Science and Technology Institute of Applied Physics and Computational Mathematics Center for Applied Physics and Technology College of Engineering Peking University

We propose two spatial second-order schemes for linear radiative transfer equations by using the idea of the unified gas kinetic scheme(UGKS) to construct the numerical boundary fluxes,and show that the proposed schemes are both positive and asymptotic preserving. The UGKS was proposed by Xu and Huang [29] for continuum and rarefied flows firstly, and was then applied to a linear radiative transfer equation by Mieussens in [20] where the asymptotic preserving property of UGKS is shown. Although it is asymptotic preserving, UGKS can not always keep the positivity of solutions. We first apply UGKS to discretize a linear radiative transfer equation to have a spatial second-order scheme. Then, by a detailed analysis of the numerical boundary fluxes, we are able to find the reasons why the positive preserving property of UGKS fails. Finally, we carefully employ a linear scaling limiter and a flux correction to make UGKS positive-preserving but still asymptotic-preserving. Consequently, we propose two spatial second-order positive and asymptotic preserving unified gas kinetic schemes for the linear radiative transfer equation, thus improving the earlier work [32] where only a first-order positive and asymptotic scheme is developed. The proposed schemes can well capture the solution of the diffusion limit equation in optically thick regions without requiring the cell size being smaller than the photon’s mean free path, while the solution in optically thin regions can also be well resolved in a natural way. To our best knowledge,this is the first time that a spatial high-order(second-order) positive and asymptotic preserving gas kinetic scheme for linear radiative transfer equations is constructed. Several numerical experiments are included to validate the spatial second-order accuracy, positive-and asymptotic-preserving properties of the proposed schemes.

关键词： Radiative transfer equation unified gas kinetic scheme discrete ordinate method second-order scheme positive preserving asymptotic preserving

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