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An accurate and efficient space-time Galerkin spectral method for the subdiffusion equation

Science China Mathematics 2024年第10期67卷 2387-2408页

作者： Wei Zeng Chuanju Xu School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we design and analyze a space-time spectral method for the subdiffusion ***, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.

关键词： subdiffusion equations variable transformation Ψ-Sobolev spaces well-posedness space-time Galerkin spectral method error estimate fast algorithm

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RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

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Communications on Applied Mathematics and Computation 2024年第3期6卷 1575-1599页

作者： Jun Zhu Chi-Wang Shu Jianxian Qiu State Key Laboratory of Mechanics and Control of Mechanical Structures and Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing University of Aeronautics and AstronauticsNanjing 210016JiangsuChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamen 361005FujianChina

In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.

关键词： RKDG method Steady-state problem Multi-resolution WENO limiter Triangular mesh Machine zero

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An Energy Stable Filtered Backward Euler Scheme for the MBE Equation with Slope Selection

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Numerical Mathematics(Theory,Methods and Applications) 2023年第1期16卷 165-181页

作者： Jiexin Wang Hong-Lin Liao Ying Zhao School of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China

As a promising strategy to adjust the order in the variable-order BDF algorithm,a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope *** temporal second-order convergence in the L^(2)norm is established under a convergence-solvability-stability(CSS)-consistent time-step *** CSS-consistent condition means that the maximum stepsize limit required for convergence is of the same order to that for solvability and stability(in certain norms)as the small interface parameterε→0^(+).Similar to the backward Euler scheme,the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels,including the volume conservation,the energy dissipation law and L^(2)norm *** tests are included to support the theoretical results.

关键词： MBE model time filter energy dissipation law error estimate

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THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA

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Acta Mathematica Scientia 2023年第4期43卷 1668-1674页

作者：吴菲王泽军陈芳启 Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China

In this paper,we study the global existence of BV solutions of the initial value problem for the isentropic p-system,where the state equation of the gas is given by P=Av^(-γ).Forγ>1,the general existence result for large initial data has not been *** using the Glimm scheme,Nishida,Smoller and Diperna successively obtained the global existence results for(γ-1)TV(v_(0)(x),u_(0)(x))being *** the present paper,by adopting a rescaling technique,we improve these results and obtain the global existence result under the condition that(γ-1)^(γ+1)(TV(v_(0)(x)))~(γ-1)(TV(u_(0)(x)))^(2) is small,which implies that,for fixedγ>1,either TV(v_(0)(x))or TV(u_(0)(x))can be arbitrarily large.

关键词： conservation laws p-system vanishing viscosity method BV solutions

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high-order finite volume methods for solving compressible multicomponent flows

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Science China Mathematics 2025年第3期68卷 729-760页

作者： Feng Zheng Jianxian Qiu School of Mathematics and Statistics Fujian Key Laboratory of Mathematical Analysis and ApplicationFujian Normal UniversityFuzhou 350117China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling&High-Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we propose a high-order finite volume method for solving multicomponent fluid problems. Our method couples the quasi-conservative form with the reconstruction of conservative variables in a characteristic manner. The source term and numerical fluxes are carefully designed to maintain the pressure and velocity equilibrium for the interface-only problem and preserve the equilibrium of physical parameters in a single-component fluid. These ingredients enable our scheme to achieve both high-order accuracy in the smooth region and the high resolution in the discontinuity region of the solution. Extensive numerical tests are performed to verify the high resolution and accuracy of the scheme.

关键词： finite volume high-order multicomponent flows

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Almost Periodic Type Group Actions on Compact Quantum Metric Spaces

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Acta Mathematica Sinica,English Series 2024年第2期40卷 568-594页

作者： Bo Tao LONG Wei WU College of Mathematics MIIT Key Laboratory of Mathematical Modelling and High Performance Computing of Air VehiclesNanjing University of Aeronautics and AstronauticsNanjing 211106P.R.China School of Mathematical Sciences Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghai 200241P.R.China

A compact quantum metric space is a complete order unit space A endowed with a Lipnorm *** give some characterizations of almost periodic type group actions on a compact quantum metric space(A,L)by means of several kinds of subsets of A,its induced equicontinuous actions on several important subsets of the dual Banach space A*,and the Lip-norm L with its induced metric space structures on the state space S(A)of A.

关键词： Lip-norm compact quantum metric space Lipschitz isomorphism almost periodicity Arzela-Ascoli theorem metric set

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An Efficient Nonlinear Multigrid Solver for the Simulation of Rarefied Gas Cavity Flow

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Communications in Computational Physics 2023年第7期34卷 357-391页

作者： Zhicheng Hu Guanghan Li School of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China

We study efficient simulation of steady state for multi-dimensional rarefied gas flow,which is modeled by the Boltzmann equation with BGK-type collision term.A nonlinear multigrid solver is proposed to resolve the efficiency issue by the following *** unified framework of numerical regularized moment method is first adopted to derive the high-quality discretization of the underlying problem.A fast sweeping iteration is introduced to solve the derived discrete problem more efficiently than the usual time-integration scheme on a single level *** it as the smoother,the nonlinear multigrid solver is then established to significantly improve the convergence *** OpenMP-based parallelization is applied in the implementation to further accelerate the *** experiments for two lid-driven cavity flows and a bottom-heated cavity flow are carried out to investigate the performance of the resulting nonlinear multigrid *** results show the wonderful efficiency and robustness of the solver for both first-and second-order spatial discretization.

关键词： Boltzmann equation moment method multigrid rarefied gas flow steady state

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Existence in the Large for Pressure-Gradient System

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Chinese Annals of Mathematics,Series B 2022年第4期43卷 509-522页

作者： Shuxin ZHANG Zejun WANG Department of Mathematics Nanjing University of Aeronautics and AstronauticsKey Laboratory of Mathematical and High Performance Computing of Air Vehicles(NUAA)MIITNanjing 210016China

In this paper,the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial *** this end,some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions(see[Diperna,R.J.,Existence in the large for quasilinear hyperbolic conservation laws,***.,52(3),1973,244–257])are *** they construct the approximate solution sequence through Glimm *** establishing accurate local interaction estimates,they prove the boundedness of the approximate solution sequence and its total variation.

关键词： Pressure-gradient system Riemann problem Diperna’s conditions Glimm scheme BV space

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An oscillation-free Hermite WENO scheme for hyperbolic conservation laws

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Science China Mathematics 2024年第2期67卷 431-454页

作者： Zhuang Zhao Jianxian Qiu School of Mathematical Sciences and Institute of Natural Sciences Shanghai Jiao Tong UniversityShanghai 200240China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, the sixth-order oscillation-free Hermite weighted essentially non-oscillatory (OFHWENO) scheme is proposed for hyperbolic conservation laws on structured meshes, where the zeroth- andfirst-order moments are the variables for the governing equations. The main difference from other HWENOschemes existing in the literature is that we add high-order numerical damping terms in the first-order momentequations to control spurious oscillations for the OF-HWENO scheme. The OF-HWENO scheme not only canachieve the designed optimal numerical order, but also can be easily implemented as we use only one set ofstencils in the reconstruction procedure and the same reconstructed polynomials are applied for the zeroth- andfirst-order moment equations. In order to obtain the adaptive order resolution when facing discontinuities, atransition polynomial is added in the reconstruction, where the associated linear weights can also be any positivenumbers as long as their summation equals one. In addition, the OF-HWENO scheme still keeps compactnessas only immediate neighbor values are needed in the space discretization. Some benchmark numerical tests areperformed to illustrate the high-order accuracy, high resolution and robustness of the proposed scheme.

关键词： Hermite WENO scheme hyperbolic conservation laws oscillation-free adaptive order discontinuous Galerkin method

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THE ANALYTIC SMOOTHING EFFECT OF LINEAR LANDAU EQUATION WITH SOFT POTENTIALS

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Acta Mathematica Scientia 2023年第6期43卷 2597-2614页

作者：李浩光徐超江 School of Mathematics and Statistics South-Central Minzu UniversityWuhan 430074China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles MIITNanjing 210016China School of Mathematics and Key Laboratory of Mathematical MIIT Nanjing University of Aeronautics and AstronauticsNanjing 210016China

In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the... 详细信息

关键词： linear Landau equation analytic smoothing effect soft potential

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