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在三维海洋模式POM基础上建立水质模型,采用中心差分格式、迎风格式以及Smolarkiewicz迎风格式离散物质输运方程.以三维理想水槽中连续源排放的浓度场预测为例,分析3种离散格式求解所得的浓度场.结果表明,3种格式的数值解与解析解的偏差均小于20%.中心差分格式会引起解的震荡,导致物质的反向输移,出现浓度负值.迎风格式能够保证浓度的正值,但该格式带来的数值耗散导致数值解与解析解偏离较大.Smolarkiewicz迎风格式在普通迎风格式基础上引入抗扩散流速,经多次叠代,能有效降低计算中的数值耗散,提高了计算精度. 相似文献
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Liu等给出的最高导数为2的双层Boussinesq水波方程具有较好的色散性和非线性,基于该方程建立了有限差分法的三维波浪数值模型。在矩形网格上对方程进行了空间离散,采用高阶导数近似方程中的时、空项,时间积分采用混合4阶Adams-Bashforth-Moulton的预报—校正格式。模拟了深水条件下的规则波传播过程,计算波面与解析结果吻合较好,反映出数值模型能很好地刻画波面过程及波面处的速度变化;在kh=2π条件下可较为准确获得沿水深分布的水平和垂向速度,这与理论分析结果一致。最后,利用数值模型计算了规则波在三维特征地形上的传播变形,数值结果和试验数据吻合较好;高阶非线性项会对波浪数值结果产生一定的影响,当波浪非线性增强,水深减少将产生更多的高次谐波。建立的双层Boussinesq模型对强非线性波浪的演化具有较好的模拟精度。 相似文献
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《海洋湖沼通报》2021,(4)
在Liu和Fang推导的双层Boussinesq方程基础上,将其简化为一层水波方程,并建立了基于混合4阶Adams-Bashforth-Moulton时间格式的立面二维数值模型;数值模拟了波浪在潜堤上的演化过程,并将数值计算结果与相关实验结果进行了对比,验证了该数值模型的正确性。进而在不同的入射波条件下,将沿着水深分布的水平速度和垂向速度的数值模拟结果与线性、二阶、三阶解析解解析结果进行综合对比。对比结果表明,在不同的无因次水深kh条件下,数值解与解析解的整体吻合程度较好。在不同的波陡H/L条件下,数值解展现了较好的非线性特征。在不同的波高水深比H/h条件下,数值解与解析解之间的整体差异较小。可以看到,该数值模型较好地模拟了波浪垂向速度场分布,体现了其优良的综合性能。 相似文献
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研究并行算法解决应用并行计算机完成规模尽可能大的偏微分方程的数值求解问题。利用Hopf-Cole变换,将一维非线性Burgers方程转化为线性扩散方程,基于第二类Saul’yev型非对称格式和Crank-Nicolson格式对扩散方程进行差分离散,建立解Burgers方程的交替分段并行差分格式,并讨论该方法的稳定性,给出了数值算例。此算法把剖分节点分成若干组,在每组上构造能够独立求解的差分方程,因此具有并行本性,适合在高性能多处理器的并行计算机上使用。数值试验的结果表明此方法是有效的,且有较高的精度。 相似文献
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建立了求解一维全非线性Green-Naghdi水波方程的中心有限体积/有限差分混合数值格式。采用结构化网格对守恒形式的控制方程进行离散和积分,界面数值通量采用有限体积法计算,剩余项则采用中心有限差分格式求解。其中,采用中心迎风有限体积格式计算控制体界面数值通量,并结合界面变量的线性重构方法,使其在空间上具有四阶精度,通过引入静压重构技术和波浪破碎指标使模型具备处理海岸水-陆动边界及波浪破碎的能力。时间积分则采用具有总时间变差减小(Total Variation Diminishing,TVD)性质的三阶龙格-库塔法进行。应用该模型对孤立波在常水深和斜坡海岸上的传播过程及规则波跨越潜堤传播的实验进行了数值模型研究,数值计算同解析解及实验数据吻合良好。 相似文献
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在他人给出的方程的基础上,通过在其动量方程中引入含4个参数的公式,推导出了加强的适合复杂地形的水波方程,新方程的色散、变浅作用以及非线性均比原来适合复杂地形的方程有了改善:色散关系式与斯托克斯线性波的Padé(4,4)阶展开式一致;变浅作用在相对水深(波数乘水深)不大于6时与解析解符合较好;非线性在相对水深不大于1.05时保持在5%的误差之内.基于该方程,在非交错网格下建立的时间差分格式为混合4阶Adams-Bashforth-Moulton的一维数值模型,并在数值计算中利用了五对角宽带解法.数值模拟了潜堤上波浪传播变形,并将数值计算结果与实验结果进行了对比,验证了该数值模型是合理的. 相似文献
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基于一维阻尼潮波传播方程解析解,从求解数值格式及Heuristic稳定性分析方面,讨论了数值解的精度、计算耗时和摩阻系数选取等问题。研究结果表明:1)Courant数小于1时,潮波方程显格式解的精度略高于隐格式解,计算耗时少于隐格式解;2)为减少计算耗时,潮波方程的隐格式解允许较大的时间步长,但解的精度有所降低,须通过减小底床摩阻系数以保证计算精度;3)隐格式解摩阻系数的选取与Courant数有关,Courant数越大,摩阻系数的选取值比实际值越小,通过理论分析结合数值试验得到了相应的关系式。这些研究结论对实际海域的潮波传播的数值模拟具有重要的应用价值。 相似文献
11.
适用于沙坝上Bragg反射的二阶Boussinesq方程数学模型及其数值验证 总被引：1，自引：1，他引：0
在二阶 Boussinesq 方程基础上,通过引入含水深导数项对该方程进行了理论上的改进,使得该方程在应用于无限沙坝 Bragg反射问题时与理论解析解在更大范围内符合.基于该改进的高阶 Boussinesq 方程,在非交错网格下建立了混合 4 阶的Adams-Bashforth- Moulton 格式的数学模型.将数值模型应用到有限个连续沙坝上波浪传播变形问题的数值模拟中,通过两点法给出数值波浪反射系数,将这些反射系数与已有的实验数据进行对比,对比表明改进后的模型计算出的反射系数与实验结果吻合更好,这验证了本文理论改进的有效性. 相似文献
12.
A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations
A numerical scheme for solving the class of extended Boussinesq equations is presented. Unlike previous schemes, where the governing equations are integrated through time using a fourth-order method, a second-order Godunov-type scheme is used thus saving storage and computational resources. The spatial derivatives are discretised using a combination of finite-volume and finite-difference methods. A fourth-order MUSCL reconstruction technique is used to compute the values at the cell interfaces for use in the local Riemann problems, whilst the bed source and dispersion terms are discretised using centred finite-differences of up to fourth-order accuracy. Numerical results show that the class of extended Boussinesq equations can be accurately solved without the need for a fourth-order time discretisation, thus improving the computational speed of Boussinesq-type numerical models. The numerical scheme has been applied to model a number of standard test cases for the extended Boussinesq equations and comparisons made to physical wave flume experiments. 相似文献
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A hybrid finite-volume and finite-difference method is proposed for numerically solving the two-dimensional (2D) extended Boussinesq equations. The governing equations are written in such a way that the convective flux is approximated using finite volume (FV) method while the remaining terms are discretized using finite difference (FD) method. Multi-stage (MUSTA) scheme, instead of commonly used HLL or Roe schemes, is adopted to evaluate the convective flux as it has the simplicity of centred scheme and accuracy of upwind scheme. The third order Runge–Kutta method is used for time marching. Wave breaking and wet–dry interface are also treated in the model. In addition to model validation, the emphasis is given to compare the merits and limitations of using MUSTA scheme and HLL scheme in the model. The analytical and experimental data available in the literature have been used for the assessment. Numerical tests demonstrate that the developed model has the advantages of stability preserving, shock-capturing and numerical efficiency when applied in the complex nearshore region. Compared with that using HLL scheme, the proposed model has comparable numerical accuracy, but requires slightly less computation time and is much simpler to code. 相似文献
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A mode-splitting method is applied to the quasi-3D nearshore circulation equations in generalized curvilinear coordinates. The gravity wave mode and the vorticity wave mode of the equations are derived using the two-step projection method. Using an implicit algorithm for the gravity mode and an explicit algorithm for the vorticity mode, we combine the two modes to derive a mixed difference–differential equation with respect to surface elevation. McKee et al.'s [McKee, S., Wall, D.P., and Wilson, S.K., 1996. An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms. J. Comput. Phys., 126, 64–76.] ADI scheme is then used to solve the parabolic-type equation in dealing with the mixed derivative and convective terms from the curvilinear coordinate transformation. Good convergence rates are found in two typical cases which represent respectively the motions dominated by the gravity mode and the vorticity mode. Time step limitations imposed by the vorticity convective Courant number in vorticity-mode-dominant cases are discussed. Model efficiency and accuracy are verified in model application to tidal current simulations in San Francisco Bight. 相似文献
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In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton's method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C~0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry. 相似文献
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W. E. Rogers J. M. Kaihatu H. A. H. Petit N. Booij L. H. Holthuijsen 《Ocean Engineering》2002,29(11):1357-1390
The numerical schemes for the geographic propagation of random, short-crested, wind-generated waves in third-generation wave models are either unconditionally stable or only conditionally stable. Having an unconditionally stable scheme gives greater freedom in choosing the time step (for given space steps). The third-generation wave model SWAN (“Simulated WAves Nearshore”, Booij et al., 1999) has been implemented with this type of scheme. This model uses a first order, upwind, implicit numerical scheme for geographic propagation. The scheme can be employed for both stationary (typically small scale) and nonstationary (i.e. time-stepping) computations. Though robust, this first order scheme is very diffusive. This degrades the accuracy of the model in a number of situations, including most model applications at larger scales. The authors reduce the diffusiveness of the model by replacing the existing numerical scheme with two alternative higher order schemes, a scheme that is intended for stationary, small-scale computations, and a scheme that is most appropriate for nonstationary computations. Examples representative of both large-scale and small-scale applications are presented. The alternative schemes are shown to be much less diffusive than the original scheme while retaining the implicit character of the particular SWAN set-up. The additional computational burden of the stationary alternative scheme is negligible, and the expense of the nonstationary alternative scheme is comparable to those used by other third generation wave models. To further accommodate large-scale applications of SWAN, the model is reformulated in terms of spherical coordinates rather than the original Cartesian coordinates. Thus the modified model can calculate wave energy propagation accurately and efficiently at any scale varying from laboratory dimensions (spatial scale O(10 m) with resolution O(0.1 m)), to near-shore coastal dimension (spatial scale O(10 km) with resolution O(100 m)) to oceanic dimensions (spatial scale O(10 000 km) with resolution O(100 km). 相似文献
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1 IntroductionThe shallow water equations (SWE) are frequent-ly used as a mathematical model for water flows incoastal areas, lakes, estuaries, etc. Thus, they are animportant tool to simulate a variety of problems relat-ed to coastal engineering, environment, ecology, etc.(Bermúdez et al., 1998). On the basis of solving theone-dimensional (1D) SWE, Hu et al. (2000) have de-veloped a model capable of simulating storm wavespropagating in the coastal surf zone and overtopping asea wall. Ano… 相似文献
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Presented here is a compact explicit difference scheme of high accuracy for solving the extended Boussinesq equations.For time discretization,a three-stage explicit Runge-Kutta method with TVD property is used at predicting stage,a cubic spline function is adopted at correcting stage,which made the time discretization accuracy up to fourth order;For spatial discretization,a three-point explicit compact difference scheme with arbitrary order accuracy is employed.The extended Boussinesq equations derived by Beji and Nadaoka are solved by the proposed scheme.The numerical results agree well with the experimental data.At the same time,the comparisons of the two numerical results between the present scheme and low accuracy difference method are made,which further show the necessity of using high accuracy scheme to solve the extended Boussinesq equations.As a valid sample,the wave propagation on the rectangular step is formulated by the present scheme,the modelled results are in better agreement with the experimental data than those of Kittitanasuan. 相似文献
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Numerical storm surge model with higher order finite difference method of lines for the coast of Bangladesh 总被引：1，自引：0，他引：1
In this study, the method of lines (MOLs) with higher order central difference approximation method coupled with the classical fourth order Runge-Kutta (RK(4,4)) method is used in solving shallow water equations (SWEs) in Cartesian coordinates to foresee water levels associated with a storm accurately along the coast of Bangladesh. In doing so, the partial derivatives of the SWEs with respect to the space variables were discretized with 5-point central difference, as a test case, to obtain a system of ordinary differential equations with time as an independent variable for every spatial grid point, which with initial conditions were solved by the RK(4,4) method. The complex land-sea interface and bottom topographic details were incorporated closely using nested schemes. The coastal and island boundaries were rectangularized through proper stair step representation, and the storing positions of the scalar and momentum variables were specified according to the rules of structured C-grid. A stable tidal regime was made over the model domain considering the effect of the major tidal constituent, M2 along the southern open boundary of the outermost parent scheme. The Meghna River fresh water discharge was taken into account for the inner most child scheme. To take into account the dynamic interaction of tide and surge, the generated tidal regime was introduced as the initial state of the sea, and the surge was then made to come over it through computer simulation. Numerical experiments were performed with the cyclone April 1991 to simulate water levels due to tide, surge, and their interaction at different stations along the coast of Bangladesh. Our computed results were found to compare reasonable well with the limited observed data obtained from Bangladesh Inland Water Transport Authority (BIWTA) and were found to be better in comparison with the results obtained through the regular finite difference method and the 3-point central difference MOLs coupled with the RK(4,4) method with regard to the root mean square error values. 相似文献
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一种基于Roe格式的有限体积法在二维溃坝问题中的应用 总被引：3，自引：0，他引：3
溃坝属于典型的非恒定含间断的浅水问题。应用有限体积法离散二维浅水控制方程的守恒型方程组,将基于近似黎曼解的Roe格式用于数值计算溃坝问题,并利用MUSCL方法构造二阶空间积分格式和预测-校正二步法构造二阶时间格式,从而使数值解的整体达到二阶,提高了精度。文中算法在一维溃坝的Stoker问题的数值结果与解析解进行对比,结果证明了此方法的可行性。应用此方法在二维溃坝问题上的结果,说明了此算法可有效模拟溃坝水流的演进过程。 相似文献