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A POD REDUCED-ORDER SPDMFE EXTRAPOLATING ALGORITHM FOR hyperbolic equations

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Acta Mathematica Scientia 2014年第3期34卷 872-890页

作者：罗振东李宏 School of Mathematics and Physics North China Electric Power University School of Mathematical Sciences Inner Mongolia University

In this article, a proper orthogonal decomposition （POD） method is used to study a classical splitting positive definite mixed finite element （SPDMFE） formulation for second- order hyperbolic equations. A POD reduced-order SPDMFE extrapolating algorithm with lower dimensions and sufficiently high accuracy is established for second-order hyperbolic equations. The error estimates between the classical SPDMFE solutions and the reduced-order SPDMFE solutions obtained from the POD reduced-order SPDMFE extrapolating algorithm are provided. The implementation for solving the POD reduced-order SPDMFE extrapolating algorithm is given. Some numerical experiments are presented illustrating that the results of numerical computation are consistent with theoretical conclusions, thus validating that the POD reduced-order SPDMFE extrapolating algorithm is feasible and efficient for solving second-order hyperbolic equations.

关键词： Proper orthogonal decomposition splitting positive definite mixed finite element formulation hyperbolic equations error estimate

ADER Methods for hyperbolic equations with a Time-Reconstruction Solver for the Generalized Riemann Problem: the Scalar Case

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Communications on Applied Mathematics and Computation 2020年第3期2卷 369-402页

作者： R.Demattè V.A.Titarev G.I.Montecinos E.F.Toro Department of Aerospace Science and Technology Politecnico di MilanoMilanItaly Dorodnicyn Computing Center Federal Research Center Computer Science and ControlRussianAcademy of SciencesMoscowRussia Departamento de Ciencias Naturales y Tecnología Universidad de AysénCoyhaiqueChile Laboratory of Applied Mathematics DICAMUniversity of TrentoTrentoItaly

The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years,including both methodological aspects as well as very ambitious *** spite of methodological progress,the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem(GRP)remain the centre of attention in the ADER *** the original formulation of ADER schemes,the proposed solution procedure for the GRP was based on(i)Taylor series expansion of the solution in time right at the element interface,(ii)subsequent application of the Cauchy-Kowalewskaya procedure to convert time derivatives to functionals of space derivatives,and(iii)solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series *** realistic problems the Cauchy-Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly *** this paper we propose a new class of solvers for the GRP that avoid the Cauchy-Kowalewskaya procedure and result in simpler ADER *** is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element *** relying on a time Taylor series expansion of the solution at the interface,the time derivatives are then easily calculated from the time-reconstruction *** resulting schemes are called ADER-TR.A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method,thus deducing linear stability ***,via careful numerical experiments,we deduce stability regions for the corresponding non-linear *** examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER ***

关键词： hyperbolic equations Finite volume ADER methods Generalized Riemann problem(GRP) Time-reconstruction(TR)

A REDUCED FE FORMULATION BASED ON POD METHOD FOR hyperbolic equations

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Acta Mathematica Scientia 2012年第5期32卷 1997-2009页

作者：罗振东欧秋兰吴加荣谢正辉 School of Mathematics and Physics North China Electric Power University LASG Institute of Atmospheric PhysicsChinese Academy of Sciences

A proper orthogonal decomposition（POD） method was successfully used in the reduced-order modeling of complex *** this paper,we extend the applications of POD method,namely,apply POD method to a classical finite element（FE） formulation for second-order hyperbolic equations with real practical applied background,establish a reduced FE formulation with lower dimensions and high enough accuracy,and provide the error estimates between the reduced FE solutions and the classical FE solutions and the implementation of algorithm for solving reduced FE formulation so as to provide scientific theoretic basis for service *** numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical ***,it is shown that the reduced FE formulation based on POD method is feasible and efficient for solving FE formulation for second-order hyperbolic equations.

关键词： proper orthogonal decomposition finite element formulation error estimate hyperbolic equations

AENO:a Novel Reconstruction Method in Conjunction with ADER Schemes for hyperbolic equations

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Communications on Applied Mathematics and Computation 2023年第2期5卷 776-852页

作者： Eleuterio F.Toro Andrea Santacá Gino I.Montecinos Morena Celant Lucas O.Müller Laboratory of Applied Mathematics DICAMUniversity of TrentoTrentoItaly Department of Mathematics University of TrentoTrentoItaly Department of Natural Sciences and Technology Universidad de AysénObispo Vielmo 62CoyhaiqueChile

In this paper,we present a novel spatial reconstruction scheme,called AENO,that results from a special averaging of the ENO polynomial and its closest neighbour,while retaining the stencil direction decided by the ENO choice.A variant of the scheme,called m-AENO,results from averaging the modified ENO(m-ENO)polynomial and its closest *** concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system,in conjunction with the fully discrete,high-order ADER approach implemented up to fifth order of accuracy in both space and *** results,as compared to the conventional ENO,m-ENO and WENO schemes,are very ***,our results show that the L_(1)-errors of the novel AENO approach are the smallest for most cases ***,for a chosen error size,AENO turns out to be the most efficient method of all five methods tested.

关键词： hyperbolic equations High-order ADER ENO/m-ENO/WENO Novel reconstruction technique AENO/m-AENO

Further Oscillation Results for a Class of hyperbolic equations with Continuous Distributed Deviating Arguments

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Chinese Quarterly Journal of Mathematics 2012年第1期27卷 145-151页

作者： ZHANG Meng SONG Guo-hua College of Forest Beijing Forestry UniversityBeijing 100083China School of Science Beijing University of Civil Engineering and ArchitectureBeijing 100044China

A class of hyperbolic equations with continuous distributed deviating arguments is considered and its oscillation theorems are *** theorems are of higher degree of generality and deal with the cases which are not cove... 详细信息

关键词： oscillation criteria hyperbolic equations continuous distributed deviating arguments

ON AN INVERSE INITIAL-BOUNDARY VALUE PROBLEM FOR A ONE-DIMENSIONAL hyperbolic SYSTEM OF DIFFERENTIAL equations

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Acta Mathematicae Applicatae Sinica 1997年第1期13卷 33-44页

作者：窦学丽朱本仁 Beijing Research Institute for Remote Sensing Information Beijing China Department of Mathematics Shandong University Jinan China

In this paper, the inverse boundary value problem of the hyperbolic system of first-order deferential equations is discussed. The estimate of the solution and the quantitative analysis about its stability are obtained... 详细信息

关键词： Inverse problem hyperbolic equations eigenvalue problem spectral function,integral kernel stability

A Fifth Order Alternative Mapped WENO Scheme for Nonlinear hyperbolic Conservation Laws

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Advances in Applied Mathematics and Mechanics 2022年第1期14卷 275-298页

作者： Uttam Singh Rajput Krishna Mohan Singh Department of Mechanical and Industrial Engineering Indian Institute of Technology RoorkeeRoorkee 247667India

In this work,we have developed a fifth-order alternative mapped weighted essentially nonoscillatory(AWENO-M)finite volume scheme using non-linear weights of mapped WENO reconstruction scheme of Henrick et al.(***.,207(2005),pp.542-567)for solving hyperbolic conservation *** reconstruction of numerical flux is done using primitive variables instead of conservative *** present scheme results in less spurious oscillations near discontinuities and shows higher-order accuracy at critical points compared to the alternative WENO scheme(AWENO)based on traditional non-linear weights of Jiang and Shu(***.,228(1996),pp.202-228).The third-order Runge-Kutta method has been used for solution advancement in *** Harten-Lax-van Leer-Contact(HLLC)shock-capturing method is used to provide necessary upwinding into the *** performance of the present scheme is evaluated in terms of accuracy,computational cost,and resolution of discontinuities by using various one and two-dimensional test cases.

关键词： High resolution scheme unsteady non-linear weights numerical fluxes alternative WENO scheme hyperbolic equations

关于一类二阶非线性双曲型方程全离散有限元方法的稳定性和收敛性估计

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高等学校计算数学学报 2002年第1期24卷 15-22页

作者：刘小华西南石油学院计算机科学系南充637001

Stability and error estimates are established for the finite element method to solve a class of second order hyperbolic ***,optimal rates of convergence in H 1 and L 2 norm are obtained.

关键词： finite element hyperbolic equations error estimate.

RECENT PROGRESS ON CONTROLLABILITY/OB SERVABILITY FOR SYSTEMS GOVERNED BY PARTIAL DIFFERENTIAL equations

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Journal of Systems Science & Complexity 2010年第3期23卷 527-545页

作者： Hongheng LI Qi LU Xu ZHANG School of Mathematics Sichuan University Chengdu 610064 China Key Laboratory of Systems and Control Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing 100190 China

The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for linear partial differential equations, including the iteration method for the null controllability of the time-invariant heat equation and the Rellich-type multiplier method for the exact controllability of the time-invariant wave equation, and especially a unified controllability/observability theory for parabolic and hyperbolic equations based on a global Carleman estimate. Then, the authors present sharp global controllability results for both semi-linear parabolic and hyperbolic equations, based on linearization approach, sharp observability estimates for the corresponding linearized systems and the fixed point argument. Finally, the authors survey the local null controllability result for a class of quasilinear parabolic equations based on the global Carleman estimate, and the local exact controllability result for general hyperbolic equations based on a new unbounded perturbation techniaue.

关键词： Controllability hyperbolic equations observability parabolic equations.