ADIF,a FORTRAN IV program for solving the shallow-water equations

A computer program is documented, implementing a linear Alternating Direction Implicit (ADI) method for a limited-area finite difference integration of the shallow-water equations on a β plane. Arbitrarily large time-steps can be used with this method, which is stable unconditionally for the linearized equations. The method also is efficient computationally, as the difference equations are factored into one-dimensional operators which approximately reproduce the original set of equations, and this obviates the necessity for solving a matrix of large bandwidth. A line-printer plot contouring the height field is generated as part of the output. Program options include the determination at each time step of two of the integral invariants of the shallow-water equation. For long-term runs either a dissipative term is provided, or a nonlinear lateral eddy viscosity coefficient of a friction force.     

Conservative semi-implicit semi-Lagrangian scheme for simulation of shallow flows

A fully conservative semi-Lagrangian (SL) scheme is presented to solve for the shallow-water equations. Existing inherently conservative SL schemes only ensure the conservation of mass while momentum is not fully conserved. The gravity terms, which are mainly responsible for the wave structure in dam break flows, are then discretized by using traditional non-conservative Eulerian schemes. In the presence of large variations in water surface (e.g., dam-break type flows), such an approach leads to incorrect shock speed and highly oscillatory results. Indeed, if the conservation of the gravity terms is forced, the use of existing schemes will be restricted to small time steps. In this paper we present a fully conservative scheme which can accurately simulate the shallow flows with a large time step. In our approach, both convective and gravitational terms are treated in a conservative manner, which ensures an accurate shock speed. The fully conservation property improves considerably the performance of common SL schemes for a wide range of practical applications.     

Coupling superposed 1D and 2D shallow-water models: Source terms in finite volume schemes

We study the superposition of 1D and 2D shallow-water equations with non-flat topographies, in the context of river-flood modeling. Since we superpose both models in the bi-dimensional areas, we focus on the definition of the coupling term required in the 1D equations. Using explicit finite volume schemes, we propose a definition of the discrete coupling term leading to schemes globally well-balanced (the global scheme preserves water at rest whatever if overflowing or not). For both equations (1D and 2D), we can consider independent finite volume schemes based on well-balanced Roe, HLL, Rusanov or other scheme, then the resulting global scheme remains well-balanced. We perform a few numerical tests showing on the one hand the well-balanced property of the resulting global numerical model, on the other hand the accuracy and robustness of our superposition approach. Therefore, the definition of the coupling term we present allows to superpose a local 2D model over a 1D main channel model, with non-flat topographies and mix incoming-outgoing lateral fluxes, using independent grids and finite volume solvers.     

Mathematical Modelling Of Wreck Events Originated By Dam Collapse

Based on Saint-Venant (shallow water) equations, in this paper the mathematical model of wreck events produced by dam collapse is constructed. A two-layer difference scheme with non-linear regularisation is used for the numerical solution of the aforementioned model. The convergence of this difference scheme in Eulerian variables with non-linear regularisation to the smooth solutions of one-dimensional Saint-Venant equations are considered for a Cauchy problem with periodic (in spatial variables) solutions. The proof of difference scheme convergence is conducted using the energetic method. The existence and uniqueness of the difference scheme solution is proved. That the difference scheme converges in mesh norm $L_2$ with speed $O\lpar h^2\rpar$ in the class of sufficiently smooth solutions of the difference scheme is also proved.     

Optimal Error Estimates of Semi-implicit Galerkin Method for Time-Dependent Nematic Liquid Crystal Flows

This paper focuses on the optimal error estimates of a linearized semi-implicit scheme for the nematic liquid crystal flows, which is used to describe the time evolution of the materials under the influence of both the flow velocity and the microscopic orientation configurations of rod-like liquid crystal flows. Optimal error estimates of the scheme are proved without any restriction of time step by using an error splitting technique proposed by Li and Sun. Numerical results are provided to confirm the theoretical analysis and the stability of the semi-implicit scheme.     

Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods

In this paper, we investigate a scheme for nonlinear reaction-diffusion equations using the mixed finite element methods. To linearize the mixed method equations, we use the two-grid algorithm. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^\frac12)H=\mathcal{O}(h^{\frac{1}{2}}). As a result, solving such a large class of nonlinear equations will not much more difficult than the solution of one linearized equation.     

On numerical stabilization in the solution of Saint-Venant equations using the finite element method

Solving the Saint-Venant equations by using numerical schemes like finite difference and finite element methods leads to some unwanted oscillations in the water surface elevation. The reason for these oscillations lies in the method used for the approximation of the nonlinear terms. One of the ways of smoothing these oscillations is by adding artificial viscosity into the scheme. In this paper, by using a suitable discretization, we first solve the one-dimensional Saint-Venant equations by a finite element method and eliminate the unwanted oscillations without using an artificial viscosity. Second, our main discussion is concentrated on numerical stabilization of the solution in detail. In fact, we first convert the systems resulting from the discretization to systems relating to just water surface elevation. Then, by using M-matrix properties, the stability of the solution is shown. Finally, two numerical examples of critical and subcritical flows are given to support our results.     

An Energy-Stable High-Order Central Difference Scheme for the Two-Dimensional Shallow Water Equations

An energy-stable high-order central finite difference scheme is derived for the two-dimensional shallow water equations. The scheme is mathematically formulated using the semi-discrete energy method for initial boundary value problems described in Olsson (1995, Math. Comput. 64, 1035–1065): after symmetrizing the equations via a change to entropy variables, the flux derivatives are entropy-split enabling the formulation of a semi-discrete energy estimate. We show experimentally that the entropy-splitting improves the stability properties of the fully discretized equations. Thus, the dependence on numerical dissipation to keep the scheme stable for long term time integrations is reduced relative to the original unsplit form, thereby decreasing non-physical damping of solutions. The numerical dissipation term used with the entropy-split equations is in a form which preserves the semi-discrete energy estimate. A random one-dimensional dam break calculation is performed showing that the shock speed is computed correctly for this particular case, however it is an open question whether the correct shock speed will be computed in generalMSC: 35Q35; 65M12; 65M06Supported in part by the New Zealand Marsden Fund, grant UOA827     

自顶向下聚集型代数多重网格预条件的健壮性与参数敏感性研究

本文针对自顶向下聚集型代数多重网格预条件,进行了健壮性与参数敏感性研究。对从各向同性与各向异性偏微分方程边值问题离散所得的多种稀疏线性方程组,首先对问题规模敏感性进行了研究,并与基于强连接的经典聚集型算法进行了系统比较,发现虽然对沿不同坐标轴具有强各向异性的问题,基于坐标分割的自顶向下聚集型算法不如基于强连接的经典聚集算法,但对其它所有情形,自顶向下聚集型算法都具有明显优势,特别是在采用Jacobi光滑时,优势更加显著。之后,对最粗网格层的分割数与每次每个子图进行分割时的分割数这两个参数进行了敏感性分析,发现在采用Jacobi光滑求解五点差分离散所得的稀疏线性方程组时,自顶向下聚集型算法对这两个参数存在敏感性,虽然大部分情形下,迭代次数比较稳定,但在少量几种情形下,迭代次数明显增加。而对从九点差分离散得到的稀疏线性方程组,以及在采用Gauss-Seidel光滑的情况下,算法对这两个参数的选取不再具有敏感性,迭代次数都比较稳定。综合分析表明,自顶向下聚集型代数多重网格预条件具有较好的健壮性,特别是在采用Gauss-Seidel光滑,或采用九点差分离散时,健壮性表现更加充分。     

An analysis of ‘ industrial dynamics ’

In this paper a linearized version of an ‘ industrial dynamics ’ simulation mode discussed by Forrester (1061) is presented and analysed. It is shown that at least for the stop change in final demand treated by Forrester (19R1) the behaviour of the linearized model presented here is not dissimilar to the behaviour of Forrester's nonlinear model. Analysis of the linearized model shows that the poor transient behaviour of the system is due primarily to the modes associated with the exponential smoothing of the demand.     

Preconditioning linearized equations

When solving systems of nonlinear equations, generally the first step is to linearize them. When using interval methods, one usually preconditions the linear equations by multiplying by a real (i.e. non-interval) matrix which approximates the inverse of the center of the interval Jacobian. In this paper, we show that it is better to precondition the original nonlinear equations rather than the linearized ones.     

Conservative upwind difference schemes for the Euler equations

In a recent paper [1] a number of numerical schemes for the shallow water equations based on a conservative linearization are analyzed. In particular, it is established that the schemes are related through the use of a source term. In this paper this technique is applied to the Euler equations, and further analysis suggests a new formulation of an existing scheme having the same key properties.     

Semiconductor device simulation using a viscous-hydrodynamic model

In this article we deal with a hydrodynamic model of Navier–Stokes (NS) type for semiconductors including a physical viscosity in the momentum and energy equations. A stabilized finite difference scheme with upwinding based on the characteristic variables is used for the discretization of the NS equations, while a mixed finite element scheme is employed for the approximation of the Poisson equation. A consistency result for the method is established showing that the scheme is first-order accurate in both space and time. We also perform a stability analysis of the numerical method applied to a linearized incompletely parabolic system in two space dimensions with vanishing viscosity. A thorough numerical parametric study as a function of the heat conductivity and of the momentum viscosity is carried out in order to investigate their effect on the development of shocks in both one and two space dimensional devices.     

A novel linearly-weighted gradient smoothing method (LWGSM) in the simulation of fluid dynamics problem

This paper describes a novel linearly-weighted gradient smoothing method (LWGSM). The proposed method is based on irregular cells and thus can be used for problems with arbitrarily complex geometrical boundaries. Based on the analysis about the compactness and the positivity of coefficients of influence of their stencils for approximating a derivative, one favorable scheme (VIII) is selected among total eight proposed discretization schemes. This scheme VIII is successively verified and carefully examined in solving Poisson equations, subjected to changes in the number of nodes, the shapes of cells and the irregularity of triangular cells, respectively. Strong form of incompressible Navier–Stokes equations enhanced with artificial compressibility terms are tackled, in which the spatial derivatives are approximated by consistent and successive use of gradient smoothing operation over smoothing domains at various locations. All the test cases using LWGSM solver exhibits its robust, stable and accurate behaviors. The attained incompressible LWGSM solutions show good agreements with experimental and literature data. Therefore, the proposed LWGSM can be reliably used for accurate solutions to versatile fluid flow problems.     

Theoretical Investigation of Time Suboptimal Control of Industrial Manipulators Along Specified Paths

A method for the time suboptimal control of an industrial manipulator from an initial position and orientation to a final position and orientation as it moves along a specified path is proposed. Nonlinear system equations that describe the manipulator motion are linearized at each time step along the path. A method which gives the control inputs (joint angular velocities) for time suboptimal control of the manipulator is developed. In the formulation, joint angular velocity and acceleration limitations are also taken into consideration. A six degree of freedom elbow type manipulator is used in numerical examples to verify the method developed.     

Numerical study of fourth-order linearized compact schemes for generalized NLS equations

The fourth-order compact approximation for the spatial second-derivative and several linearized approaches, including the time-lagging method of Zhang et al. (1995), the local-extrapolation technique of Chang et al. (1999) and the recent scheme of Dahlby et al. (2009), are considered in constructing fourth-order linearized compact difference (FLCD) schemes for generalized NLS equations. By applying a new time-lagging linearized approach, we propose a symmetric fourth-order linearized compact difference (SFLCD) scheme, which is shown to be more robust in long-time simulations of plane wave, breather, periodic traveling-wave and solitary wave solutions. Numerical experiments suggest that the SFLCD scheme is a little more accurate than some other FLCD schemes and the split-step compact difference scheme of Dehghan and Taleei (2010). Compared with the time-splitting pseudospectral method of Bao et al. (2003), our SFLCD method is more suitable for oscillating solutions or the problems with a rapidly varying potential.     

On the use of symmetrizing variables for vacuums

The paper is devoted to the computation of shallow-water equations (or Euler equations) in using an approximate Godunov scheme called VFRoe, when the flow may include dry areas (or very low density regions). This is achieved with the help of symmetrizing variables. Overall we are able to insure the discrete preservation of positive variables on interfaces, and at the same time to compute vacuum occurence or propagation of shock waves over a near-vacuum. A short section is also dedicated to the non-conservative hyperbolic equations arising within the setting of one-equation or two-equation turbulent compressible models. Many numerical tests confirm the capabilities of the scheme, and measuring the L¹ error norm in particular cases enables us to specify the actual rate of convergence.     

Flooding and Drying in Discontinuous Galerkin Finite-Element Discretizations of Shallow-Water Equations. Part 1: One Dimension

Free boundaries in shallow-water equations demarcate the time-dependent water line between ‘‘flooded’’ and ‘‘dry’’ regions. We present a novel numerical algorithm to treat flooding and drying in a formally second-order explicit space discontinuous Galerkin finite-element discretization of the one-dimensional or symmetric shallow-water equations. The algorithm uses fixed Eulerian flooded elements and a mixed Eulerian–Lagrangian element at each free boundary. When the time step is suitably restricted, we show that the mean water depth is positive. This time-step restriction is based on an analysis of the discretized continuity equation while using the HLLC flux. The algorithm and its implementation are tested in comparison with a large and relevant suite of known exact solutions. The essence of the flooding and drying algorithm pivots around the analysis of a continuity equation with a fluid velocity and a pseudodensity (in the shallow water case the depth). It therefore also applies, for example, to space discontinuous Galerkin finite-element discretizations of the compressible Euler equations in which vacuum regions emerge, in analogy of the above dry regions. We believe that the approach presented can be extended to finite-volume discretizations with similar mean level and slope reconstruction.This revised version was published online in July 2005 with corrected volume and issue numbers.     

摄影测量共线方程的单位四元数描述

为探讨四元数在摄影测量共线方程严密解算中的应用问题,从四元数的基本理论出发,详细推导以单位四元数矩阵表达的共线方程的严密线性化公式,该线性化公式无须对旋转矩阵进行微分。以单像后方交会和光束法平差为例采用模拟数据和实际数据实验。结果表明,推导的以四元数矩阵为基础的线性化共线方程具有形式简单、初值无关性和收敛速度快的优点。具有一定的实用价值。     

迎风紧致格式与热流计算

１．引 言 众所周知,ＴＶＤ格式是能够高质量地捕捉激波的方法,但在计算粘性绕流时许多ＴＶＤ格式数值耗散太大,不能正确模拟粘性流动,因而无法正确计算热流值．文献［３］指出,采用高精度格式可适当放松对网格雷诺数的要求,因此发展三阶或三阶以上的格式是需要的．文献［４］研究了迎风紧致群速度控制格式（ＵＣＧＶＣ格式）在 Ｅｕｌｅｒ方程中的应用,提高了对激波的分辨率,优于通常二阶精度ＴＶＤ格式．本文在文献［４］的基础上给出了利用迎风紧致格式求解ＮＳ方程．它是ＵＣＧＶＣ格式在粘性流计算中的推广．对于方程中的无粘…