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A Physics-Informed Structure-Preserving Numerical Scheme for the Phase-Field Hydrodynamic Model of Ternary Fluid Flows

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Numerical Mathematics(Theory,Methods and Applications) 2023年第3期16卷 565-596页

作者： Qi Hong Yuezheng Gong Jia Zhao School of Mathematics Nanjing University of Aeronautics and Astronautics/Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA)MIITNanjing 211106China Department of Mathematics and Statistics Utah State UniversityLoganUT84322USA

Phase-field models are widely used in studying multiphase flow *** the complexity and strong nonlinearity,designing accurate,efficient,and stable numerical algorithms to solve these models has been an active research field for *** paper proposes a novel numerical scheme to solve a highly cited and used phase field hydrodynamic model for simulating ternary phase fluid *** main novelty is the introduction of a supplementary variable to reformulate the original problem into a constrained optimization *** reformulation leads to several advantages for our proposed numerical algorithms compared with many existing numerical techniques for solving this ***,the developed schemes allow more straightforward calculations for the hydrodynamic phase-field models by solving a few decoupled Helmholtz or Poisson-type systems with a constant precomputable coefficient matrix,remarkably reducing the computational ***,the numerical schemes can maintain mass conservation and energy dissipation at the discrete ***,the developed scheme based on the secondorder backward difference formula respects the original energy dissipation law that differs from many existing schemes,such as the IEQ,SAV,and Lagrange multiplier approaches for which a modified energy dissipation law is ***,rigorous proof of energy stability and practical implementation strategies are *** conduct adequate 2D and 3D numerical tests to demonstrate the proposed schemes’accuracy and effectiveness.

关键词： Energy stable phase field Cahn-Hilliard-Navier-Stokes supplementary variable method

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A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

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Communications on Applied Mathematics and Computation 2023年第1期5卷 403-427页

作者： Liang Li Jun Zhu Chi-Wang Shu Yong-Tao Zhang Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing University of Aeronautics and AstronauticsNanjing 210016JiangsuChina 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 Department of Applied and Computational Mathematics and Statistics University of Notre DameNotre DameIN 46556USA

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local ***,they are easy to be applied to a general hyperbolic *** deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the *** this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing *** experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the *** results show highorder accuracy and good performance of the ***,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state *** examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

关键词： Fixed-point fast sweeping methods Multi-resolution WENO schemes Steady state ILW procedure Convergence

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EXISTENCE OF PERIODIC SOLUTIONS TO AN ISOTHERMAL RELATIVISTIC EULER SYSTEM

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Acta Mathematica Scientia 2022年第1期42卷 155-171页

作者： Fei WU Zejun WANG Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MHTNanjing 211106China

In this paper,we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV ***,we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant *** on these properties,we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme,and prove that there exists an entropy solution V(x,t)which belongs to L^(∞)∩BV_(loc)(R×R_(+)).

关键词： isothermal relativistic Euler system Glimm scheme Riemann problem BV space periodicity

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Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

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Communications on Applied Mathematics and Computation 2024年第1期6卷 605-624页

作者： Feng Zheng Jianxian Qiu College of Mathematics and Statistics Fujian Normal UniversityFuzhou350117FujianChina School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling&High-Performance Scientific Computing Xiamen UniversityXiamen361005FujianChina

In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation *** can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary ***,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO *** numerical tests are performed to verify the high resolution and high accuracy of the scheme.

关键词： Finite volume Dimension by dimension HWENO Hyperbolic conservation laws

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A high ORDER SCHEME FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE CAPUTO-HADAMARD DERIVATIVE

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Journal of Computational Mathematics 2025年第3期43卷 615-640页

作者： Xingyang Ye Junying Cao Chuanju Xu School of Science Jimei UniversityXiamen 361021China School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper,we consider numerical solutions of the fractional diffusion equation with theαorder time fractional derivative defined in the Caputo-Hadamard sense.A high order time-stepping scheme is constructed,analyzed,and numerically *** contribution of the paper is twofold:1)regularity of the solution to the underlying equation is investigated,2)a rigorous stability and convergence analysis for the proposed scheme is performed,which shows that the proposed scheme is 3+αorder *** numerical examples are provided to verify the theoretical statement.

关键词： Caputo-Hadamard derivative Fractional differential equations high order scheme Stability and convergence analysis

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Traveling wave solutions for a three-component noncooperative systems arising in nonlocal diffusive biological models

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International Journal of Biomathematics 2024年第7期17卷 187-216页

作者： Ran Zhang Hongyong Zhao School of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 211106 P.R.China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA)MIIT Nanjing 211106P.R.China School of Mathematical Science Heilongjiang University Harbin 150080P.R.China

This paper aims to study the existence of traveling wave solutions(TWS)for a three-component noncooperative systems with nonlocal *** main results reveal that when a threshold R>1,there exists a critical wave speed c^(*)>*** using sub-and super-solution methods and Schauder's fixed point theorem,we prove that the system admits a nontrivial TWS for each c≥c^(*).Meanwhile,we show that there exists no nontrivial TWS for c

关键词： Traveling wave solution nonlocal diffusion biological model Schauder's fixed point theorem noncooperative systems

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Derivatives of Intersection Local Time for Two Independent Symmetricα-stable Processes

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Acta Mathematica Sinica,English Series 2024年第5期40卷 1273-1292页

作者： Huan ZHOU Guang Jun SHEN Qian YU Department of Mathematics Anhui Normal UniversityWuhu 241002P.R.China Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106P.R.China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106P.R.China

In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient condition for the existence of the derivatives,which makes us obtain the exponential integrability and H?lder *** we show that this condition is also necessary for the existence of derivatives of intersection local time at the ***,we also study the power variation of the derivatives.

关键词： Symmetric stable processes intersection local time exponential integrability power variation

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A Novel Spectral Method for the Subdiffusion Equation 13th

A Novel Spectral Method for the Subdiffusion Equation

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13th International Conference on Spectral and high Order Methods, ICOSAHOM 2021

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

ISBN: (纸本)9783031204319

In this paper, we design and analyze a novel spectral method for the subdiffusion equation. As it has been known, the solutions of this equation are usually singular near the initial time. Consequently, direct application of the traditional high-order numerical methods is inefficient. We try to overcome this difficulty in a novel approach by combining variable transformation techniques with spectral methods. The idea is to first use suitable variable transformation to re-scale the underlying equation, then construct spectral methods for the re-scaled equation. We establish a new variational framework based on the ψ-fractional Sobolev spaces. This allows us to prove the well-posedness of the associated variational problem. The proposed spectral method is based on the variational problem and generalized Jacobi polynomials to approximate the re-scaled fractional differential equation. Our theoretical and numerical investigation show that the proposed method is exponentially convergent for general right hand side functions, even though the exact solution has very limited regularity. Implementation details are also provided, along with a series of numerical examples to show the efficiency of the proposed method. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Sobolev spaces

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MONOTONICITY CORRECTIONS FOR NINE-POINT SCHEME OF DIFFUSION EQUATIONS

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Journal of Computational Mathematics 2024年第5期42卷 1305-1327页

作者： Wang Kong Zhenying Hong Guangwei Yuan Zhiqiang Sheng 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 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 present a nonlinear correction technique to modify the nine-point scheme proposed in[SIAM ***.,30:3(2008),1341-1361]such that the resulted scheme preserves the *** first express the flux by the cell-centered unknowns and edge unknowns based on the stencil of the nine-point ***,we use a nonlinear combination technique to get a monotone *** order to obtain a cell-centered finite volume scheme,we need to use the cell-centered unknowns to locally approximate the auxiliary *** present a new method to approximate the auxiliary unknowns by using the idea of an improved multi-points flux *** numerical results show that the new proposed scheme is robust,can handle some distorted grids that some existing finite volume schemes could not handle,and has higher numerical accuracy than some existing positivity-preserving finite volume schemes.

关键词： Monotonicity corrections Diffusion equation Improved MPFA Distorted meshes

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A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

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Numerical Mathematics(Theory,Methods and Applications) 2023年第1期16卷 111-139页

作者： Min Zhang Weizhang Huang Jianxian Qiu School of Mathematical Sciences Peking UniversityBeijing 100871China Department of Mathematics University of KansasLawrenceKansas 66045USA School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical ModelingHigh-Performance Scientific ComputingXiamen UniversityXiamenFujian 361005China

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin(DG)solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping.A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy(CFL)conditions established for fixed and uniform *** this work,we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations.A stability analysis is presented for a moving mesh DG method for linear scalar conservation *** on the analysis,a new selection strategy of the time step is proposed,which takes into consideration the coupling of theα-function(that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity)and the heights of the mesh *** analysis also suggests several stable combinations of the choices of theα-function in the numerical scheme and in the time step *** results obtained with a moving mesh DG method for Burgers’and Euler equations are *** comparison purpose,numerical results obtained with an error-based time step-size selection strategy are also given。

关键词： Discontinuous Galerkin method adaptive mesh moving mesh CFL condition stability

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