INITIAL BOUNDARY VALUE PROBLEMS FOR A CLASS OF NONLINEAR INTEGRO－PARTIAL DIFFERENTIAL EQUATIONS

ＩＮＩＴＩＡＬＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＩＮＴＥＧＲＯ－ＰＡＲＴＩＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＣｕｉＳｈａｎｇ－ｂｉｎ（崔尚斌）（Ｄｅｐｔ．ｏｆＭａｔｈ．,ＬａｎｚｈｏｕＵｎｉ...     

FINITE ELEMENTAN ALYSIS FOR THE UNSTEADY NEARSHORECIR CULATION DUE TO WAVE－CURRENT INTER－ACTION（I）──NUMERICAL MODEL

ＦＩＮＩＴＥＥＬＥＭＥＮＴＡＮＡＬＹＳＩＳＦＯＲＴＨＥＵＮＳＴＥＡＤＹＮＥＡＲＳＨＯＲＥＣＩＲＣＵＬＡＴＩＯＮＤＵＥＴＯＷＡＶＥ－ＣＵＲＲＥＮＴＩＮＴＥＲ－ＡＣＴＩＯＮ（Ｉ）——ＮＵＭＥＲＩＣＡＬＭＯＤＥＬＷｕＷｅｉ－ｘｉｏｎｇ（吴伟雄）（Ｔｏｎｇｊ...     

THE CALCULATION OF THE MULTIPLY－CONNECTED ELASTIC PLANE PROBLEMS BY MEANS OF STRESS FUNCTIONS OF MULTIPLE COMPLEX VARIABLES

ＴＨＥＣＡＬＣＵＬＡＴＩＯＮＯＦＴＨＥＭＵＬＴＩＰＬＹ－ＣＯＮＮＥＣＴＥＤＥＬＡＳＴＩＣＰＬＡＮＥＰＲＯＢＬＥＭＳＢＹＭＥＡＮＳＯＦＳＴＲＥＳＳＦＵＮＣＴＩＯＮＳＯＦＭＵＬＴＩＰＬＥＣＯＭＰＬＥＸＶＡＲＩＡＢＬＥＳｗａｎｇＬｉｎｄｉａｎｇ（王林江）Ｌ...     

THREE DIMENSIONAL STRESS ANALYSIS OF SYMMETRIC COMPOSITE LAMINATES UNDER UNIAXIAL EXTENSION AND IN－PLANE PURE SHEAR

ＴＨＲＥＥＤＩＭＥＮＳＩＯＮＡＬＳＴＲＥＳＳＡＮＡＬＹＳＩＳＯＦＳＹＭＭＥＴＲＩＣＣＯＭＰＯＳＩＴＥＬＡＭＩＮＡＴＥＳＵＮＤＥＲＵＮＩＡＸＩＡＬＥＸＴＥＮＳＩＯＮＡＮＤＩＮ－ＰＬＡＮＥＰＵＲＥＳＨＥＡＲＣｈｉｃｎｗｅｉ－ｚａｎｇ（钱伟长）,Ｈｕａｎｇ...     

Numerical Simulation for Evolutionary History of Three-Dimensional Basin

ＮＵＭＥＲＩＣＡＬＳＩＭＵＬＡＴＩＯＮＦＯＲＥＶＯＬＵＴＩＯＮＡＲＹＨＩＳＴＯＲＹＯＦＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＢＡＳＩＮＹｕａｎＹｉ－ｒａｎｇ（袁益让）；ＷａｎｇＷｅｎ－ｑｉａ（王文洽）；ＹａｎｇＤａｎ－ｐｉｎｇ（羊丹平）（Ｄｅｐａｒｔｍｅ...     

EAPPROXIMATE INERTIAL MANIFOLDS FOR THE SYSTEM OF THE J-J EQUATIONS

ＡＰＰＲＯＸＩＭＡＴＥＩＮＥＲＴＩＡＬＭＡＮＩＦＯＬＤＳＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＪ－ＪＥＱＵＡＴＩＯＮＳＡＰＰＲＯＸＩＭＡＴＥＩＮＥＲＴＩＡＬＭＡＮＩＦＯＬＤＳＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＪ－ＪＥＱＵＡＴＩＯＮＳ￥ＣａｉＲｉｚｅｎｇ（...     

THE ANALYTICAL STUDY ON THE LASER INDUCED REVERSE－PLUGGING EFFECT BY USING THE CLASSICAL ELASTIC PLATE THEORY（II）──REVERSE－BULGE MOTION

ＴＨＥＡＮＡＬＹＴＩＣＡＬＳＴＵＤＹＯＮＴＨＥＬＡＳＥＲＩＮＤＵＣＥＤＲＥＶＥＲＳＥ－ＰＬＵＧＧＩＮＧＥＦＦＥＣＴＢＹＵＳＩＮＧＴＨＥＣＬＡＳＳＩＣＡＬＥＬＡＳＴＩＣＰＬＡＴＥＴＨＥＯＲＹ（ＩＩ）──ＲＥＶＥＲＳＥ－ＢＵＬＧＥＭＯＴＩＯＮ￥(周益春，...     

THE APPLICATION OF MULTI－SCALE PERTURBATION METHOD TO THE STABILITY ANALYSIS OF PLANE COUETTE FLOW

ＴＨＥＡＰＰＬＩＣＡＴＩＯＮＯＦＭＵＬＴＩ－ＳＣＡＬＥＰＥＲＴＵＲＢＡＴＩＯＮＭＥＴＨＯＤＴＯＴＨＥＳＴＡＢＩＬＩＴＹＡＮＡＬＹＳＩＳＯＦＰＬＡＮＥＣＯＵＥＴＴＥＦＬＯＷＺｈｏｕＺｈｅ－ｗｅｉ（周哲玮）（ＳｈａｎｇｈａｉＵｎｉｖｅｒｓｉｌｙ;Ｓｈａｇ...     

THE ANALYTICAL STUDY ON THE LASER INDUCED REVERSE－PLUGGING EFFECT BY USING THE CLASSICAL ELASTIC PLATE THEORY（Ⅰ）－TEMPERATURE FIELDS

ＴＨＥＡＮＡＬＹＴＩＣＡＬＳＴＵＤＹＯＮＴＨＥＬＡＳＥＲＩＮＤＵＣＥＤＲＥＶＥＲＳＥ－ＰＬＵＧＧＩＮＧＥＦＦＥＣＴＢＹＵＳＩＮＧＴＨＥＣＬＡＳＳＩＣＡＬＥＬＡＳＴＩＣＰＬＡＴＥＴＨＥＯＲＹ（Ⅰ）－ＴＥＭＰＥＲＡＴＵＲＥＦＩＥＬＤＳＺｈｏｕＹｉｃｈｕｎ...     

THE EQUATION OF MOTION FOR THE SYSTEM OF THE VARIABLE MASS IN THE NON－LINEAR NON－HOLONOMIC SPACE

ＴＨＥＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＶＡＲＩＡＢＬＥ　ＡＳＳＩＮＴＨＥＮＯＮ－ＬＩＮＥＡＲＮＯＮ－ＨＯＬＯＮＯＭＩＣＳＰＡＣＥＴＨＥＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＶＡＲＩＡＢＬＥ...     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient high‐resolution relaxation schemes for hyperbolic systems of conservation laws

In this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ? ? [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

A preconditioning technique for steady Euler solutions

This paper presents a preconditioning technique for solving a two‐dimensional system of hyperbolic equations. The main attractive feature of this approach is that, unlike a technique based on simply extending the solver for a one‐dimensional hyperbolic system, convergence and stability analysis can be investigated. This method represents a genuine numerical algorithm for multi‐dimensional hyperbolic systems. In order to demonstrate the effectiveness of this approach, applications to solving a two‐dimensional system of Euler equations in supersonic flows are reported. It is shown that the Lax–Friedrichs scheme diverges when applied to the original Euler equations. However, convergence is achieved when the same numerical scheme is employed using the same CFL number to solve the equivalent preconditioned Euler system. Copyright © 1999 John Wiley & Sons, Ltd.     

非稳态Hamilton-Jacobi方程的7阶加权紧致非线性格式

Hamilton-Jacobi (HJ) 方程是一类重要的非线性偏微分方程, 在物理学、流体力学、图像处理、微分几何、金融数学、最优化控制理论等方面有着广泛的应用. 由于HJ方程的弱解存在但不唯一, 且解的导数可能出现间断, 导致其数值求解具有一定的难度. 本文提出了非稳态HJ方程的7阶精度加权紧致非线性格式 (WCNS). 该格式结合了Hamilton函数的Lax-Friedrichs型通量分裂方法和一阶空间导数左、右极限值的高阶精度混合节点和半节点型中心差分格式. 基于7点全局模板和4个4点子模板推导了半节点函数值的高阶线性逼近和4个低阶线性逼近, 以及全局模板和子模板的光滑度量指标. 为避免间断附近数值解产生非物理振荡以及提高格式稳定性, 采用WENO型非线性插值方法计算半节点函数值. 时间离散采用3阶TVD型Runge-Kutta方法. 通过理论分析验证了WCNS格式对于光滑解具有最佳的7阶精度. 为方便比较, 经典的7阶WENO格式也被推广用于求解HJ方程. 数值结果表明, 本文提出的WCNS格式能够很好地模拟HJ方程的精确解, 且在光滑区域能够达到7阶精度; 与经典的同阶WENO格式相比, WCNS格式在精度、收敛性和分辨率方面更优, 计算效率略高.      

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

A COMPUTATIONAL METHOD FOR THE NAVIER-STOKES EQUATIONS AT ALL SPEEDS

ＩｎｔｒｏｄｕｃｔｉｏｎＦｏｒｃｏｍｐｕｔａｔｉｏｎｏｆｔｈｅｖｉｓｃｏｕｓｃｏｍｐｒｅｓｓｉｂｌｅｆｌｏｗｓｔｈｅＮａｖｉｅｒ＿Ｓｔｏｋｅｓｅｑｕａｔｉｏｎｓａｒｅｇｅｎｅｒａｌｌｙｒｅｐｒｅｓｅｎｔｅｄｉｎｔｈｅｃｏｎｓｅｒｖａｔｉｏｎｌａｗｆｏｒｍａｓａｈｙｐｅｒｂｏｌｉｃｓｙｓｔｅｍ .Ｌａｃｋｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｔｏｏｌｔｏａｎａｌｙｚｅｔｈｉｓｎｏｎｌｉｎｅａｒｓｙｓｔｅｍ ,ｔｈｅｎｕｍｅｒｉｃａｌｍｅｔｈｏｄｓｕｓｅｄｉｎｓｏｌｖｉｎｇｔｈｅｎｏｎｌｉｎｅａｒｈｙｐｅ…     

A C2‐continuous high‐resolution upwind convection scheme

A bounded upwinding scheme for numerical solution of hyperbolic conservation laws and Navier–Stokes equations is presented. The scheme is based on convection boundedness criterion and total variation diminishing stability criteria and developed by employing continuously differentiable functions. The accuracy of the scheme is verified by assessing the error and observed convergence rate on 1‐D benchmark test cases. A comparative study between the new scheme and conventional total variation diminishing/convection boundedness criterion‐based upwind schemes to solve standard nonlinear hyperbolic conservation laws is also accomplished. The scheme is then examined in the simulation of Newtonian and non‐Newtonian fluid flows of increasing complexity; a satisfactory agreement has been observed in terms of the overall behavior. Finally, the scheme is used to study the hydrodynamics of a gas‐solid flow in a bubbling fluidized bed. Copyright © 2013 John Wiley & Sons, Ltd.     

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.     

COMPUTATION OF TRANSIENT NONLINEAR SHIP WAVES USING AN ADAPTIVE ALGORITHM

An indirect boundary integral method is used to solve transient nonlinear ship wave problems. A resulting mixed boundary value problem is solved at each time-step using a mixed Eulerian– Lagrangian time integration technique. Two dynamic node allocation techniques, which basically distribute nodes on an ever changing body surface, are presented. Both two-sided hyperbolic tangent and variational grid generation algorithms are developed and compared on station curves. A ship hull form is generated in parametric space using a B-spline surface representation. Two-sided hyperbolic tangent and variational adaptive curve grid-generation methods are then applied on the hull station curves to generate effective node placement. The numerical algorithm, in the first method, used two stretching parameters. In the second method, a conservative form of the parametric variational Euler–Lagrange equations is used the perform an adaptive gridding on each station. The resulting unsymmetrical influence coefficient matrix is solved using both a restarted version of GMRES based on the modified Gram–Schmidt procedure and a line Jacobi method based on LU decomposition. The convergence rates of both matrix iteration techniques are improved with specially devised preconditioners. Numerical examples of node placements on typical hull cross-sections using both techniques are discussed and fully nonlinear ship wave patterns and wave resistance computations are presented.     

A high order spectral volume solution to the Burgers' equation using the Hopf–Cole transformation

A limiter free high order spectral volume (SV) formulation is proposed in this paper to solve the Burgers' equation. This approach uses the Hopf–Cole transformation, which maps the Burgers' equation to a linear diffusion equation. This diffusion equation is solved in an SV setting. The local discontinuous Galerkin (LDG) and the LDG2 viscous flux discretization methods were employed. An inverse transformation was used to obtain the numerical solution to the Burgers' equation. This procedure has two advantages: (i) the shock can be captured, without the use of a limiter; and (ii) the effects of SV partitioning becomes almost redundant as the transformed equation is not hyperbolic. Numerical studies were performed to verify. These studies also demonstrated (i) high order accuracy of the scheme even for very low viscosity; (ii) superiority of the LDG2 scheme, when compared with the LDG scheme. In general, the numerical results are very promising and indicate that this procedure can be applied for obtaining high order numerical solutions to other nonlinear partial differential equations (for instance, the Korteweg–de Vries equations) which generate discontinuous solutions. Copyright © 2011 John Wiley & Sons, Ltd.