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Science China Mathematics 2022年第8期65卷 1731-1748页

作者： Waixiang Cao Dongfang Li Zhimin Zhang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

In this paper,we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave *** error estimates in time and superconvergent error estimates in space are established without certain time-step *** key is to estimate directly the solution bounds in the H-norm for both the nonlinear wave equation and the corresponding fully discrete scheme,while the previous investigations rely on the temporal-spatial error splitting *** examples are presented to confirm energy-conserving properties,unconditional convergence and optimal error estimates,respectively,of the proposed fully discrete schemes.

关键词： scalar auxiliary variable wave equations stability error estimate superconvergence

Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation

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Science China Mathematics 2022年第10期65卷 2201-2218页

作者： Xiaoli Li Jie Shen School of Mathematics Shandong UniversityJinan 250100China Department of Mathematics Purdue UniversityWest LafayetteIN 47907USA

In this paper,we construct efficient schemes based on the scalar auxiliary variable block-centered finite difference method for the modified phase field crystal equation,which is a sixth-order nonlinear damped wave *** schemes are linear,conserve mass and unconditionally dissipate a pseudo *** prove rigorously second-order error estimates in both time and space for the phase field variable in discrete *** also present some numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy.

关键词： modified phase field crystal scalar auxiliary variable energy stability error estimate numerical experiments

Novel Partitioned Time-Stepping Algorithms for Fast Computation of Random Interface-Coupled Problems with Uncertain Parameters

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Numerical Mathematics(Theory,Methods and Applications) 2024年第1期17卷 145-180页

作者： Yizhong Sun Jiangshan Wang Haibiao Zheng School of Mathematical Sciences East China Normal UniversityShanghaiChina School of Mathematical Sciences Ministry of Education Key Laboratory of Mathematics and Engineering ApplicationsShanghai Key Laboratory of PMMPEast China Normal UniversityShanghaiChina

The simulation of multi-domain,multi-physics mathematical models with uncertain parameters can be quite demanding in terms of algorithm design and com-putation *** main objective in this paper is to examine a physical interface coupling between two random dissipative systems with uncertain *** to the complexity and uncertainty inherent in such interface-coupled problems,un-certain diffusion coefﬁcients or friction parameters often arise,leading to consid-ering random *** employ Monte Carlo methods to produce independent and identically distributed deterministic heat-heat model samples to address ran-dom systems,and adroitly integrate the ensemble idea to facilitate the fast calcu-lation of these *** achieve unconditional stability,we introduce the scalar auxiliary variable(SAV)method to overcome the time constraints of the ensemble implicit-explicit ***,for a more accurate and stable scheme,the ensemble data-passing algorithm is raised,which is unconditionally stable and convergent without any auxiliary *** algorithms employ the same co-efﬁcient matrix for multiple linear systems and enable easy parallelization,which can signiﬁcantly reduce the computational ***,numerical experiments are conducted to support the theoretical results and showcase the unique features of the proposed algorithms.

关键词： scalar auxiliary variable ensemble algorithm random interface-coupled problems implicit-explicit partitioned method data-passing partitioned method

A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrodinger Equation

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Advances in Applied Mathematics and Mechanics 2023年第3期15卷 583-601页

作者： Jun Yang Nianyu Yi School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

*** this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear Schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear Schrodinger equation into an equivalent system and to transform the energy into a quadratic *** use the standard continuous finite element method for the spatial discretization,and the relaxation Runge-Kutta method for the time *** mass and energy conservation laws are shown for the semi-discrete finite element scheme,and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta *** examples are presented to demonstrate the accuracy of the proposed method,and the conservation of mass and energy in long time simulations.

关键词： Schrodinger equation mass conservation energy conservation finite element method relaxation Runge-Kutta scalar auxiliary variable

An SAV Method for Imaginary Time Gradient Flow Model in Density Functional Theory

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Advances in Applied Mathematics and Mechanics 2023年第3期15卷 719-736页

作者： Ting Wang Jie Zhou Guanghui Hu Department of Mathematics Faculty of Science and TechnologyUniversity of MacaoMacao SARChina School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China Zhuhai UM Science&Technology Research Institute ZhuhaiGuangdong 519030China Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications University of MacaoMacao SARChina

In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure *** handle the orthonormality constraint on those wave functions,two kinds of penalty terms are introduced in designing the modified energy functional in SAV,i.e.,one for the norm preserving of each wave function,another for the orthogonality between each pair of different wave functions.A numerical method consisting of a designed scheme and a linear finite element method is used for the ***,the desired unconditional decay of modified energy can be obtained from our method,while computationally,both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical *** importantly,numerical results show that the orthonormality among those wave functions can be automatically preserved,without explicitly preserving orthogonalization *** implies the potential of our method in large-scale simulations in density functional theory.

关键词： Density functional theory gradient flow scalar auxiliary variable unconditional energy stability orthonormalization free

SAV Finite Element Method for the Peng-Robinson Equation of State with Dynamic Boundary Conditions

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Advances in Applied Mathematics and Mechanics 2023年第1期15卷 139-158页

作者： Changhui Yao Zhaoyue Du Lei Yang School of Mathematics and Statistics Zhengzhou UniversityZhengzhouHenan 450001China Faculty of Information Technology Macao University of Science and TechnologyMacao

In this paper,the Peng-Robinson equation of state with dynamic boundary conditions is discussed,which considers the interactions with solid *** first,the model is introduced and the regularization method on the nonlinear term is ***,The scalar auxiliary variable(SAV)method in temporal and finite element method in spatial are used to handle the Peng-Robinson equation of ***,the energy dissipation law of the numerical method is ***,we acquire the convergence of the discrete SAV finite element method(FEM).Finally,a numerical example is provided to confirm the theoretical result.

关键词： Peng-Robinson equation of state dynamic boundary conditions scalar auxiliary variable finite element method error estimates