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Numerical simulation of a complex moving rigid body under the impingement of a shock wave in 3D

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Advances in Aerodynamics 2022年第1期4卷 147-175页

作者： Shihao Liu Ziqiang Cheng Yan Jiang Jianfang Lu Mengping Zhang Shuhai Zhang School of Mathematical Sciences University of Science and Technology of ChinaHefei230026AnhuiChina School of Mathematics Hefei University of TechnologyHefei230000AnhuiChina School of Mathematics South China University of TechnologyGuangzhou510641GuangdongChina State Key Laboratory of Aerodynamics China Aerodynamics Research and Development CenterMianyang621000SichuanChina

In this paper,we take a numerical simulation of a complex moving rigid body under the impingement of a shock wave in three-dimensional space.Both compressible inviscid fluid and viscous fluid are considered with suitable boundary conditions.We develop a high order numerical boundary treatment for the complex moving geometries based on finite difference methods on fixed Cartesian meshes.The method is an extension of the inverse Lax-Wendroff(ILW)procedure in our works(Cheng et al.,Appl Math Mech(Engl Ed)42:841-854,2021;Liu et al.)for 2D problems.Different from the 2D case,the local coordinate rotation in 3D required in the ILW procedure is not unique.We give a theoretical analysis to show that the boundary treatment is independent of the choice of the rotation,ensuring the method is feasible and valid.Both translation and rotation of the body are taken into account in this paper.In particular,we reformulate the material derivative for inviscid fluid on the moving boundary with no-penetration condition,which plays a key role in the proposed algorithm.Numerical simulations on the cylinder and sphere are given,demonstrating the good performance of our numerical boundary treatments.

关键词： Inverse Lax-Wendroff procedure high order accuracy Complex moving boundary treatment Compressible inviscid fluid Compressible viscous fluid 3D

high order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

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Communications on Applied Mathematics and Computation 2023年第1期5卷 485-528页

作者： Zepeng Liu Yan Jiang Mengping Zhang Qingyuan Liu School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026AnhuiChina School of Mathematics and Physics Anhui Jianzhu UniversityHefei 230026AnhuiChina

A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The exact C-property is investigated,and comparison with the standard finite difference WENO scheme is made.Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems.The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems,indicating smaller errors compared with the Lax-Friedrichs solver.In addition,we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

关键词： Shallow water equation Well-balanced high order accuracy WENO scheme Curvilinear meshes Positivity-preserving limiter

high order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

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Communications in Computational Physics 2022年第3期31卷 913-946页

作者： Jianfang Lin Yan Xu Huiwen Xue Xinghui Zhong School of Mathematical Sciences Zhejiang UniversityHangzhouZhejiang 310058P.R.China School of Mathematical Sciences University of Science and Technology of ChinaHefeiAnhui 230026P.R.China

In this paper,we develop twofinite difference weighted essentially non-oscillatory(WENO)schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi(DP)andµ-Degasperis-Procesi(µDP)equations,which contain nonlinear high order derivatives,and possibly peakon solutions or shock waves.By introducing auxiliary variable(s),we rewrite the DP equation as a hyperbolic-elliptic system,and theµDP equation as afirst order system.Then we choose a linearfinite difference scheme with suitable order of accuracy for the auxiliary variable(s),and twofinite difference WENO schemes with unequal-sized sub-stencils for the primal variable.One WENO scheme uses one large stencil and several smaller stencils,and the other WENO scheme is based on the multi-resolution framework which uses a se-ries of unequal-sized hierarchical central stencils.Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil,both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights.Another advantage is that thefinal reconstructed polynomial on the target cell is a polynomial of the same de-gree as the polynomial over the big stencil,while the classicalfinite difference WENO reconstruction can only be obtained for specific points inside the target interval.Nu-merical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.

关键词： high order accuracy weighted essentially non-oscillatory schemes Degasperis-Procesi equation µ-Degasperis-Procesi equation finite difference method multi-resolution

Inverse Lax-Wendroff Boundary Treatment: a Survey

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Communications in Mathematical Research 2022年第3期38卷 333-350页

作者： Chi-Wang Shu Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA.

The inverse Lax-Wendroff(ILW)procedure is a numerical boundary treatment technique,which allows finite difference schemes and other schemes to achieve stability and high order accuracy when using cartesian meshes to solve boundary value problems defined on complex computational domain.In this short survey we summarize the main ingredients of the ILW procedure,discuss its applicability and stability properties,and provide possible directions of its future development.

关键词： Inverse Lax-Wendroff procedure boundary treatment high order accuracy stability complexgeometry

ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT

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Journal of Computational Mathematics 2006年第3期24卷 239-251页

作者： Zhengfu Xu Chi-Wang Shu Department of Mathematics Pennsylvania State University University Park PA 16802 USA Division of Applied Mathematics Brown University Providence Rhode Island 02912 USA

In this paper we further explore and apply our recent anti-diffusive flux corrected high order finite difference WENO schemes for conservation laws [18] to compute the SaintVenant system of shallow water equations with pollutant propagation, which is described by a transport equation. The motivation is that the high order anti-diffusive WENO scheme for conservation laws produces sharp resolution of contact discontinuities while keeping high order accuracy for the approximation in the smooth region of the solution. The application of the anti-diffusive high order WENO scheme to the Saint-Venant system of shallow water equations with transport of pollutant achieves high resolution

关键词： Anti-diffusive flux correction Sharpening contact discontinuity high order accuracy Finite difference WENO scheme Saint-Venant system of shallow water Transport of pollutant.

A high order boundary scheme to simulate complex moving rigid body under impingement of shock wave

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Applied Mathematics and Mechanics(English Edition) 2021年第6期42卷 841-854页

作者： Ziqiang CHENG Shibao LIU Yan JIANG Jianfang LU Mengping ZHANG Shuhai ZHANG School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631Chins State Key Laboratory of Aerodynamics China Aerodynamics Research and Development CenterMianyang 621000Sichuan ProvinceChina

In the paper, we study a high order numerical boundary scheme for solving the complex moving boundary problem on a fixed Cartesian mesh, and numerically investigate the moving rigid body with the complex boundary under the impingement of an inviscid shock wave. Based on the high order inverse Lax-Wendroff(ILW) procedure developed in the previous work(TAN, S. and SHU, C. W. A high order moving boundary treatment for compressible inviscid flows. Journal of Computational Physics, 230(15),6023–6036(2011)), in which the authors only considered the translation of the rigid body,we consider both translation and rotation of the body in this paper. In particular, we reformulate the material derivative on the moving boundary with no-penetration condition, and the newly obtained formula plays a key role in the proposed algorithm. Several numerical examples, including cylinder, elliptic cylinder, and NACA0012 airfoil, are given to indicate the effectiveness and robustness of the present method.

关键词： inverse Lax-Wendroff(ILW)procedure complex moving boundary scheme Cartesian mesh high order accuracy compressible inviscid shock wave

high-order discontinuous Galerkin solver on hybrid anisotropic meshes for laminar and turbulent simulations

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Applied Mathematics and Mechanics(English Edition) 2014年第7期35卷 799-812页

作者：姜振华阎超于剑 College of Aeronautics Science and Engineering Beijing University of Aeronautics and Astronautics

Efficient and robust solution strategies are developed for discontinuous Galerkin （DG） discretization of the Navier-Stokes （NS） and Reynolds-averaged NS （RANS） equations on structured/unstructured hybrid meshes. A novel line-implicit scheme is devised and implemented to reduce the memory gain and improve the computational eificiency for highly anisotropic meshes. A simple and effective technique to use the mod- ified Baldwin-Lomax （BL） model on the unstructured meshes for the DC methods is proposed. The compact Hermite weighted essentially non-oscillatory （HWENO） limiters are also investigated for the hybrid meshes to treat solution discontinuities. A variety of compressible viscous flows are performed to examine the capability of the present high- order DG solver. Numerical results indicate that the designed line-implicit algorithms exhibit weak dependence on the cell aspect-ratio as well as the discretization order. The accuracy and robustness of the proposed approaches are demonstrated by capturing com- plex flow structures and giving reliable predictions of benchmark turbulent problems.

关键词： discontinuous Galerkin （DG） method implicit method Baldwin-Lomax（BL） model high order accuracy structured/unstructured hybrid mesh

high order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

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Communications on Applied Mathematics and Computation 2021年第4期3卷 701-718页

作者： Giacomo Albi Lorenzo Pareschi Computer Science Department University of VeronaVerona 37134Italy Mathematics and Computer Science Department University of FerraraFerrara 44121Italy

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.

关键词： Semi-implicit methods Implicit-explicit methods Multistep methods Strong stability preserving high order accuracy

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE WENO SCHEMES BASED ON DIFFERENT NUMERICAL FLUXES FOR THE SHALLOW WATER EQUATIONS

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Journal of Computational Mathematics 2010年第6期28卷 807-825页

作者： Changna Lu Jianxian Qiu Ruyun Wang College of Mathematics and Physics Nanjing University of Information Science and Technology Nanjing 210044 China Department of Mathematics Nanjing University Nanjing 210093 China College of Ocean Hohai University Nanjing Jiangsu 210098 P.R. China

In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the property, and resolution of discontinuities. issues of CPU cost, accuracy, non-oscillatory

关键词： Numerical flux WENO finite volume scheme Shallow water equations high order accuracy Approximate Riemann solver Runge-Kutta time discretization.