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

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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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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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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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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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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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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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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