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A New Fifth-Order Finite Volume Central WENO Scheme for Hyperbolic Conservation Laws on Staggered Meshes

Advances in Applied Mathematics and Mechanics 2022年第5期14卷 1059-1086页

作者： Shengzhu Cui Zhanjing Tao Jun Zhu 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 AstronauticsNanjingJiangsu 210016China School of Mathematics Jilin UniversityChangchunJilin 130012China

In this paper,a new fifth-order finite volume central weighted essentially non-oscillatory(CWENO)scheme is proposed for solving hyperbolic conservation laws on staggered *** high-order spatial reconstruction procedure using a convex combination of a fourth degree polynomial with two linear polynomials(in one dimension)or four linear polynomials(in two dimensions)in a traditional WENO fashion and a time discretization method using the natural continuous extension(NCE)of the Runge-Kutta method are applied to design this new fifth-order CWENO *** new finite volume CWENO scheme uses the information defined on the same largest spatial stencil as that of the same order classical CWENO schemes[37,46]with the application of smaller number of unequal-sized spatial *** the new nonlinear weights are adopted,the new finite volume CWENO scheme could obtain the same order of accuracy and get smaller truncation errors in L1 and L¥norms in smooth regions,and control the spurious oscillations near strong shocks or contact *** new CWENO scheme has advantages over the classical CWENO schemes[37,46]on staggered meshes in its simplicity and easy extension to *** one-dimensional and two-dimensional benchmark numerical examples are provided to illustrate the good performance of this new fifthorder finite volume CWENO scheme.

关键词： Finite volume scheme central WENO scheme NCE of Runge-Kutta method staggered mesh

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DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

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Journal of Computational Mathematics 2023年第2期41卷 325-344页

作者： Hong-lin Liao Tao Tang Tao Zhou 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 Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

This is one of our series works on discrete energy analysis of the variable-step BDF *** this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM ***.,58:2294-2314]and[***.,90:1207-1226]for our previous works on the BDF2 *** this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation ***-robust stability and convergence analysis in the L^(2) norm are then *** the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step *** also present numerical tests to support our theoretical results.

关键词： Diffusion equations Variable-step third-order BDF scheme Discrete gradient structure Discrete orthogonal convolution kernels Stability and convergence

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An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media

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Annals of Applied Mathematics 2024年第4期40卷 346-392页

作者： Huangxin Chen Zixuan Wang School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamenFujian 361005China

We develop a locally mass-conservative enriched Petrov-Galerkin(EPG)method without any penalty term for the simulation of Darcy flow in fractured porous *** discrete fracture model is applied to model the fractures as the lower dimensional fracture *** new method enriches the approximation trial space of the conforming continuous Galerkin(CG)method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions in the fractures and the surrounding porous *** propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG *** solution of the pressure can be decoupled into two steps with a standard CG method and a post-processing *** post-processing correction based on the bubble functions in the matrix and the fractures can be solved separately,which is useful for parallel *** derive a priori and a posteriori error estimates for the *** examples are presented to illustrate the performance of the proposed method.

关键词： Discrete fracture model enriched Petrov-Galerkin method local mass conservation post-processing error analysis

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Extremum-Preserving Correction of the Nine-Point Scheme for Diffusion Equation on Distorted Meshes

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Advances in Applied Mathematics and Mechanics 2023年第2期15卷 402-427页

作者： Wang Kong Zhenying Hong Guangwei Yuan Zhiqiang Sheng Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjingJiangsu 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjingJiangsu 211106China Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088China HEDPS Center for Applied Physics and TechnologyPeking UniversityBeijing 100871China

In this paper,we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted *** introduce a new nonlinear approach to construct the conservative flux,that is,a linear second order flux is firstly given and a nonlinear conservative flux is then constructed by using an adaptive method and a nonlinear weighted *** new scheme does not need to use the convex combination of the cell-center unknowns to approximate the auxiliary unknowns,so it can deal with the problem with general discontinuous *** results show that our new scheme performs more robust than some existing schemes on highly distorted meshes.

关键词： Extremum-preserving correction diffusion equation distorted mesh nine-point scheme

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Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 876-902页

作者： Xuejuan Chen Zhiping Mao George Em Karniadakis School of Science Jimei UniversityXiamen 361021China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC *** then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff ***,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational ***,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.

关键词： Stiff problem generalized minimal residual geometric mesh refinement long time evolution fractional phase field models

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Local Dynamics Analysis of a Four-Dimensional Hyperchaotic Lorenz System

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Journal of Applied Mathematics and Physics 2023年第10期11卷 2789-2802页

作者： Yu Ren Liangqiang Zhou School of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA) Nanjing China

The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stability under different parameter conditions are analyzed by applying Routh-Hurwitz criterion. The results indicate that the system may exist one, three and five equilibrium points for different system parameters. Based on the central manifold theorem and normal form theorem, the pitchfork bifurcation and Hopf bifurcation are studied respectively. By using the Hopf bifurcation theorem and calculating the first Lyapunov coefficient, the Hopf bifurcation of this system is obtained as supercritical for certain parameters. Finally, the results of theoretical parts are verified by some numerical simulations.

关键词： Hyperchaotic Lorenz System Stability Pitchfork Bifurcation Hopf Bifurcation Central Manifold

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high-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

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Communications in Computational Physics 2023年第5期33卷 1217-1239页

作者： Yizhou Lu Jun Zhu Shengzhu Cui Zhenming Wang Linlin Tian Ning Zhao State Key Laboratory of Mechanics and Control for Aerospace Structures Nanjing University of Aeronautics and AstronauticsJiangsu 210000P.R.China Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles Nanjing University of Aeronautics and AstronauticsJiangsu 210000P.R.China Jiangsu Key Laboratory of Hi-Tech Research for Wind Turbine Design Nanjing University of Aeronautics and AstronauticsJiangsu 210000P.R.China College of Aerospace Engineering Nanjing University of Aeronautics and AstronauticsJiangsu 210000P.R.China

In this paper,a new multi-resolution weighted essentially non-oscillatory(MR-WENO)limiter for high-order local discontinuous Galerkin(LDG)method is designed for solving Navier-Stokes equations on triangular *** MR-WENO limiter is a new extension of the finite volume MR-WENO *** new limiter uses information of the LDG solution essentially only within the troubled cell itself,to build a sequence of hierarchical L^(2)projection polynomials from zeroth degree to the highest degree of the *** an example,a third-order LDGmethod with associated same orderMR-WENO limiter has been developed in this paper,which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact *** linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is *** is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled *** MR-WENO limiter is very simple to construct,and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured *** spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular *** classical viscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.

关键词： Local discontinuous Galerkin method multi-resolution WENO limiter triangular meshes Navier-Stokes equations

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The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 1128-1146页

作者： Hong-Lin Liao Xiaohan Zhu Jindi Wang MIIT Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles Nanjing 211106P.R.China School of Mathematics Nanjing University of Aeronautics and Astronautics211106P.R.China School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105P.R.China

In this work,we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s *** L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools,i.e.,the discrete orthogonal convolution kernels and discrete complementary convolution *** the discrete embedding techniques and the fractional Gronwall inequality are applied to establish an L^(2)norm error estimate on nonuniform time *** adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

关键词： Time-fractional Allen-Cahn equation adaptive L1 scheme variational energy dissipation law orthogonal convolution kernels complementary convolution kernels

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high Order Conservative Finite Difference/Fourier Spectral Methods for Inviscid Surface Quasi-Geostrophic Flows

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Communications in Computational Physics 2022年第10期32卷 1474-1509页

作者： Nan Zhang Zhiping Mao Tao Xiong School of Mathematical Sciences Xiamen UniversityXiamenFujian361005P.R.China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenFujian361005P.R.China

In this paper,we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasigeostrophic equations,which include a spectral fractional Laplacian determining the vorticity for the transport velocity of the potential *** fractional Laplacian is approximated by a Fourier-Galerkin spectral method,while the time evolution of the potential temperature is discretized by a high order conservative finite difference *** essentially non-oscillatory(WENO)reconstructions are also considered for *** to a low regularity of problems involving such a fractional Laplacian,especially in the critical or supercritical regime,directly applying the Fourier spectral method leads to a very oscillatory transport velocity associated with the gradient of the vorticity,*** smooth *** of using an artificial filter,we propose to reconstruct the velocity from the vorticity with central difference *** results are performed to demonstrate the good performance of our proposed approach.

关键词： Finite difference WENO Fourier spectral fractional Laplacian surface quasigeostrophic flow

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A high Order Bound Preserving Finite Difference Linear Scheme for Incompressible Flows

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Communications in Computational Physics 2022年第6期32卷 126-155页

作者： Guoliang Zhang Tao Xiong School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

We propose a high order finite difference linear scheme combined with ahigh order bound preserving maximum-principle-preserving (MPP) flux limiter tosolve the incompressible flow system. For such problem with highly oscillatory structure but not strong shocks, our approach seems to be less dissipative and much lesscostly than a WENO type scheme, and has high resolution due to a Hermite reconstruction. Spurious numerical oscillations can be controlled by the weak MPP fluxlimiter. Numerical tests are performed for the Vlasov-Poisson system, the 2D guidingcenter model and the incompressible Euler system. The comparison between the linearand WENO type schemes, with and without the MPP flux limiter, will demonstrate thegood performance of our proposed approach.

关键词： Finite difference scheme high order Hermite reconstruction MPP flux limiter incompressible flow Vlasov-Poisson

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