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Robust a Simulation for Shallow Flows with Friction on Rough Topography

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Numerical Mathematics(Theory,Methods and Applications) 2013年第2期6卷 384-407页

作者： Jian Deng Ruo Li Tao Sun Shuonan Wu School of Mathematical Sciences Peking UniversityBeijing 100871China HEDPS&capt LMAM&School of Mathematical SciencesPeking UniversityBeijing 100871China Reservoir Characterization Division ExxonMobil Upstream Research Company P.O.Box 2189 HoustonTexas 77252USA

In this paper,we propose a robust finite volume scheme to numerically solve the shallow water equations on complex rough *** major difficulty of this problem is introduced by the stiff friction force term and the wet/dry interface *** analytical integration method is presented for the friction force term to remove the *** the vicinity of wet/dry interface,the numerical stability can be attained by introducing an empirical parameter,the water depth tolerance,as extensively adopted in *** propose a problem independent formulation for this parameter,which provides a stable scheme and preserves the overall truncation error of δ(Δx^(3)).The method is applied to solve problems with complex rough topography,coupled with h-adaptive mesh techniques to demonstrate its robustness and efficiency.

关键词： Shallow water free interface Manning force

Well-posedness of a non-local abstract Cauchy problem with a singular integral

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Frontiers of Mathematics in China 2019年第1期14卷 77-93页

作者： Haiyan JIANG Tiao LU Xiangjiang ZHU Introduction We consider the following initial value problem for the unknown functions c(t) and z): 1 School of Mathematical Sciences Beijing Institute of Technology Beijing 100081 China School of Mathematical Sciences Peking University Beijing 100871 China capt HEDPS LMAM IFSA Collaborative Innovation Center of MoE Peking University Beijing 100871 China

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with in ow boundary conditions.

关键词： Partial integro-differential equation (PIDE) singular integral wellposedness Wigner equation

A Multi-Mesh Adaptive Finite Element Approximation to Phase Field Models

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Communications in Computational Physics 2009年第5期5卷 1012-1029页

作者： Xianliang Hu Ruo Li Tao Tang Department of Mathematics Zhejiang UniversityHangzhou 31027China Department of Applied and Computational Mathematics California Institute of TechnologyPasadenaCA 91125USA capt LMAM&School of Mathematical SciencesPeking UniversityBeijing 100871China Department of Mathematics Hong Kong Baptist UniversityKowloon TongKowloonHong Kong

In this work,we propose an efficient multi-mesh adaptive finite element method for simulating the dendritic growth in two-and *** governing equations used are the phase field model,where the regularity behaviors of the relevant dependent variables,namely the thermal field function and the phase field function,can be very *** enhance the computational efficiency,we approximate these variables on different h-adaptive *** coupled terms in the system are calculated based on the implementation of the multi-mesh h-adaptive algorithm proposed by Li(***.,pp.321-341,24(2005)).It is illustrated numerically that the multi-mesh technique is useful in solving phase field models and can save storage and the CPU time significantly.

关键词： Multi-mesh local refinement adaptive finite element phase field

Numerical Validation for High Order Hyperbolic Moment System of Wigner Equation

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Communications in Computational Physics 2014年第3期15卷 569-595页

作者： Ruo Li Tiao Lu Yanli Wang Wenqi Yao School of Mathematical Sciences Peking UniversityBeijing 100871P.R.China HEDPS and lmam Peking UniversityBeijing 100871P.R.China BICMR Peking UniversityBeijing 100871P.R.China capt Peking UniversityBeijing 100871P.R.China

A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in[6].For numerically solving the high order hyperbolic moment system therein,we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in[8],to validate the moment system of the Wigner *** method developed can keep both mass and momentum conserved,and the variation of the total energy under control though it is not strictly *** systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion,and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity *** numerical results indicate that the high order moment system in[6]is a valid model for the Wigner equation,and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.

关键词： Wigner equation NRxx method moment method

High order moment closure for Vlasov-Maxwell equations

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Frontiers of Mathematics in China 2015年第5期10卷 1087-1100页

作者： Yana DI Zhenzhong KOU Ruo LI The State Key Laboratory of Scientific and Engineering Computing Chinese Academy of Sciences Beijing 100190 China National Center for Mathematics and Interdisciplinary Sciences Chinese Academy of Sciences Beijing 100190 China Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190 China HEDPS & capt LMAM & School of Mathematical Sciences Peking University Beijing 100871 China

The extended magnetohydrodynamic models are derived based on the moment closure of the Vlasov-Maxwell （VM） equations. We adopt the Grad type moment expansion which was firstly proposed for the Boltzmann equation. A new regularization method for the Grad＇s moment system was recently proposed to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. For the VM equations, the moment expansion of the convection term is exactly the same as that in the Boltzmann equation, thus the new developed regularization applies. The moment expansion of the electromagnetic force term in the VM equations turns out to be a linear source term, which can preserve the conservative properties of the distribution function in the VM equations perfectly.

关键词： Moment closure Vlasov-Maxwell (VM) equations Boltzmannequation extended magnetohydrodynamics

An Arbitrary Order Reconstructed Discontinuous Approximation to Biharmonic Interface Problem

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Annals of Applied Mathematics 2023年第2期39卷 149-180页

作者： Yan Chen Ruo Li Qicheng Liu School of Mathematical Sciences Peking UniversityBeijing 100871China capt LMAM and School of Mathematical SciencesPeking UniversityBeijing 100871China

We present an arbitrary order discontinuous Galerkin finite element method for solving the biharmonic interface problem on the unfitted *** approximation space is constructed by a patch reconstruction process with at most one degrees of freedom per *** discrete problem is based on the symmetric interior penalty method and the jump conditions are weakly imposed by the Nitsche's *** C^(2)-smooth interface is allowed to intersect elements in a very general fashion and the stability near the interface is naturally ensured by the patch *** prove the optimal a priori error estimate under the energy norm and the L^(2) *** results are provided to verify the theoretical analysis.

关键词： Biharmonic interface problem patch reconstruction discontinuous Galerkin method

Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis:1-D Lattice Model System

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Communications in Computational Physics 2013年第7期14卷 276-300页

作者： Tiao Lu Wei Cai Jianguo Xin Yinglong Guo HEDPS&capt LMAM&School of Mathematical SciencesPeking UniversityBeijing 100871P.R.China INS Shanghai Jiaotong UniversityShanghai 200240P.R.China Department of Mathematics and Statistics University of North Carolina at CharlotteCharlotteNC 28223-0001USA School of Mathematical Sciences Peking UniversityBeijing 100871P.R.China

In the first of a series of papers,wewill study a discontinuous Galerkin(DG)framework for many electron quantum *** salient feature of this framework is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed piecewise polynomial basis in representing the Hamiltonian of the many body *** a flexibility is made possible by using the discontinuous Galerkin method to approximate the Hamiltonian matrix elements with proper constructions of numerical DG fluxes at the finite element *** this paper,we will apply the DG method to the density matrix minimization formulation,a popular approach in the density functional theory of many body Schrodinger *** density matrix minimization is to find the minima of the total energy,expressed as a functional of the density matrixρ(r,r′),approximated by the proposed enriched basis,together with two constraints of idempotency and electric *** idempotency will be handled with theMcWeeny’s purification while the neutrality is enforced by imposing the number of electrons with a penalty method.A conjugate gradient method(a Polak-Ribiere variant)is used to solve the minimization ***,the linear-scaling algorithm and the advantage of using the local orbital enriched finite element basis in the DG approximations are verified by studying examples of one dimensional lattice model systems.

关键词： Density functional theory density matrix minimization discontinuous Galerkin method linear scaling method

A DISCONTINUOUS GALERKIN METHOD BY PATCH RECONSTRUCTION FOR BIHARMONIC PROBLEM

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Journal of Computational Mathematics 2019年第4期37卷 524-540页

作者： Ruo Li Pingbing Ming Zhiyuan Sun Fanyi Yang Zhijian Yang capt LMAM and School of Mathematical Sciences Peking University Beijing 100871 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing 100190 China School of Mathematical Sciences University of Chinese Academy of Sciences No. 19A Yu-Quan Road Beijing 100049 China School of Mathematical Sciences Peking University Beijing 100871 China School of Mathematics and. Statistics Wuhan University Wuhan 430072 China

We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes.

关键词： Least-squares problem Reconstructed basis function Discontinuous Galerkin method Biharmonic problem