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AN EFFICIENT ADER DISCONTINUOUS GALERKIN SCHEME FOR DIRECTLY SOLVING hamilton-jacobi equation

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Journal of Computational Mathematics 2020年第1期38卷 58-83页

作者： Junming Duan Huazhong Tang HEDPS CAPT&LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871China School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105China

This paper proposes an efficient ADER(Arbitrary DERivatives in space and time)discontinuous Galerkin(DG)scheme to directly solve the hamilton-jacobi *** multi-stage Runge-Kutta methods used in the Runge-Kutta DG(RKDG)schemes,the ADER scheme is one-stage in time discretization,which is desirable in many *** ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and *** such predictor step,a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in *** resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG *** scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly *** explicit formulae of the scheme at third order is provided on two-dimensional structured *** computational complexity of the ADER-DG scheme is compared to that of the RKDG *** experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

关键词： hamilton-jacobi equation ADER Discontinuous Galerkin methods Local continuous spacetime Galerkin predictor High order accuracy

Finite Volume Hermite WENO Schemes for Solving the hamilton-jacobi equation

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Communications in Computational Physics 2014年第4期15卷 959-980页

作者： Jun Zhu Jianxian Qiu College of Science Nanjing University of Aeronautics and AstronauticsNanjing 210016JiangsuChina School of Mathematical Sciences Xiamen UniversityXiamen 361005FujianChina.

In this paper,we present a new type of Hermite weighted essentially nonoscillatory(HWENO)schemes for solving the hamilton-jacobi equations on the finite volume *** cell averages of the function and its first one(in one dimension)or two(in two dimensions)derivative values are together evolved via time approaching and used in the *** the major advantages of the new HWENO schemes are their compactness in the spacial field,purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial *** numerical tests are performed to illustrate the capability of the methodologies.

关键词： HWENO scheme finite volume hamilton-jacobi equation

hamilton-jacobi equationS FOR A REGULAR CONTROLLED hamiltonIAN SYSTEM AND ITS REDUCED SYSTEMS

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Acta Mathematica Scientia 2023年第2期43卷 855-906页

作者：王红 School of Mathematical Sciences and LPMC Nankai UniversityTianjin 300071China

In this paper,we give the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled hamiltonian(RCH)system and its regular reduced systems,which are called the Type I and Type II hamilton-jacobi ***,we prove two types of hamilton-jacobi theorems for an RCH system on the cotangent bundle of a configuration manifold by using the canonical symplectic form and its dynamical vector ***,we generalize the above results for a regular reducible RCH system with symmetry and a momentum map,and derive precisely two types of hamilton-jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH ***,we prove that the RCH-equivalence for the RCH system,and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries,leave the solutions of corresponding hamilton-jacobi equations ***,as an application of the theoretical results,we show the Type I and Type II hamilton-jacobi equations for the Rp-reduced controlled rigid body-rotor system and the Rp-reduced controlled heavy top-rotor system on the generalizations of the rotation group SO(3)and the Euclidean group SE(3),*** work reveals the deeply internal relationships of the geometrical structures of phase spaces,the dynamical vector fields and the controls of the RCH system.

关键词： regular controlled hamiltonian system hamilton-jacobi equation regular point reduction regular orbit reduction RCH-equivalence

Does There Exist the Applicability Limit of PDE to Describe Physical Phenomena?—A Personal Survey of Quantization, QED, Turbulence

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World Journal of Mechanics 2024年第6期14卷 97-142页

作者： Atsushi Inoue Department of Mathematics Tokyo Institute of Technology Tokyo Japan

What does it mean to study PDE (Partial Differential equation)? How and what to do “to claim proudly that I’m studying a certain PDE”? Newton mechanic uses mainly ODE (Ordinary Differential equation) and describes nicely movements of Sun, Moon and Earth etc. Now, so-called quantum phenomenum is described by, say Schrödinger equation, PDE which explains both wave and particle characters after quantization of ODE. The coupled Maxwell-Dirac equation is also “quantized” and QED (Quantum Electro-Dynamics) theory is invented by physicists. Though it is said this QED gives very good coincidence between theoretical1 and experimental observed quantities, but what is the equation corresponding to QED? Or, is it possible to describe QED by “equation” in naive sense?

关键词： Superspace Grassmann Variables hamilton-jacobi equation Quantization

High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static hamilton-jacobi equations

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Communications in Computational Physics 2022年第1期31卷 154-187页

作者： Yupeng Ren Yulong Xing Jianxian Qiu School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China Department of Mathematics The Ohio State UniversityColumbusOH 43210USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the statichamilton-jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function φ.They are now computed from the current value of φ and the previous spatial derivatives of φ. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear hamiltonian, therefore any hamiltonjacobi equations with complex hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameterε used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods.

关键词： Hermite method Weighted Essentially Non-Oscillatory(WENO)method fixedpoint iteration hamilton-jacobi equation hybrid strategy

A PDE Approach to the Long-Time Asymptotic Solutions of Contact hamilton-jacobi equations

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Wuhan University Journal of Natural Sciences 2022年第3期27卷 189-194页

作者： WANG Yujie LI Xia School of Mathematics Science Suzhou University of Science and TechnologySuzhou 215009JiangsuChina

We study the long-time asymptotic behaviour of viscosity solutions u(x,t)of the hamilton-jacobi equation u_(t)(x,t)+H(x,u(x,t),Du(x,t))=0 in T^(n)×(0,∞)with a PDE approach,where H=H(x,u,p)is coercive in p,non-decreasing in u and strictly convex in(u,p),and establish the uniform convergence of u(x,t)to an asymptotic solution u_(∞)(x)as t→∞.Moreover,u_(∞) is a viscosity solution of hamilton-jacobi equation H(x,u(x),Du(x))=0.

关键词： asymptotic solution hamilton-jacobi equation PDE approach

Symmetric Reduction and hamilton-jacobi equations of the Controlled Underwater Vehicle-Rotor System

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Communications in Mathematical Research 2023年第4期39卷 575-644页

作者： Hong Wang School of Mathematical Sciences and LPMC Nankai UniversityTianjin 300071P.R.China.

.As an application of the theoretical results,in this paper,we study the symmetric reduction and hamilton-jacobi theory for the underwater ve-hicle with two internal rotors as a regular point reducible RCH system,in the cases of coincident and non-coincident centers of the buoyancy and the *** first,we give the regular point reduction and the two types of hamilton-jacobi equations for a regular controlled hamiltonian(RCH)system with sym-metry and a momentum map on the generalization of a semidirect product Lie ***,we derive precisely the geometric constraint conditions of the reduced symplectic forms for the dynamical vector fields of the regular point reducible controlled underwater vehicle-rotor system,that is,the two types of hamilton-jacobi equations for the reduced controlled underwater vehicle-rotor system,by calculations in *** work reveal the deeply internal relationships of the geometrical structures of the phase spaces,the dynamical vector fields and the controls of the system.

关键词： Underwater vehicle with internal rotors regular controlled hamiltonian system coincident and non-coincident centers regular point reduction hamilton-jacobi equation

HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR hamilton-jacobi equationS

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Journal of Computational Mathematics 2007年第2期25卷 131-144页

作者： Jianxian Qiu Department of Mathematics Nanjing University Nanjing 210093 China

In this paper, we use Hermite weighted essentially non-oscillatory （HWENO） schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve hamilton-jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes （HWENO-RK） of Qiu and Shu [19] for hamilton-jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.

关键词： WENO scheme Hermite interpolation hamilton-jacobi equation Lax-Wendroff type time discretization High order accuracy.

A TVD Type Wavelet-Galerkin Method for hamilton-jacobi equations

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Acta Mathematicae Applicatae Sinica 2007年第2期23卷 303-310页

作者： Ling-yan Tang Song-he Song Department of Mathematics and System Science Science School National University of Defense Technology Hunan 410073 China

In this paper, we use Daubechies scaling functions as test functions for the Galerkin method, and discuss Wavelet-Galerkin solutions for the hamilton-jacobi equations. It can be proved that the schemes are TVD schemes. Numerical tests indicate that the schemes are suitable for the hamilton-jacobi equations. Furthermore, they have high-order accuracy in smooth regions and good resolution of singularities.

关键词： hamilton-jacobi equation wavelet-galerkin method daubechies wavelet TVD method