Efficient higher-order finite-difference schemes for parabolic models

Implicit finite-difference schemes for use in parabolic equation models are developed. Like the familiar Crank-Nicolson scheme, which has hitherto been used almost exclusively for the solution of these equations, these schemes are unconditionally stable and use a computational molecule of only six points on two “time” levels. However, they are accurate to a higher order than the Crank-Nicolson scheme, thus allowing the solution grid to be coarser and the solution time to be (approximately) halved. Examples of computations on constant depth are shown, in which significant reductions in time and grid-point density are achieved, for two different parabolic models. The schemes are then extended to refraction and diffraction, and are shown to have a similar effect in this more general case too. It is recommended that finite-difference schemes based on these higher-order (or Hermitian) methods replace the more commonly used Crank-Nicolson scheme in all physical domain parabolic equation models, but especially in minimax (wide-angle) equation models.     

Multilevel hydrothermodynamic model of a deep basin

A hydrothermodynamic model based on the traditional system of differential equations is discussed. The model uses conservative finite-difference schemes based on the methods of identifying the barotropic and baroclinic velocity components and on the complete inversion of the dynamic operator. Test computations for the Black Sea basin have been conducted.Translated by Vladimir A. Puchkin.     

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.     

The symmetry and stability of unstructured mesh C-grid shallow water models under the influence of Coriolis

The symmetry and stability properties of two unstructured C-grid discretisations of the shallow water equations are discussed. We establish that a scheme in which the circumcentres of the mesh triangles are used as the surface elevation points has advantageous symmetry properties and derive a Coriolis discretisation which preserves these properties. It is shown that the resulting scheme is conservative in a discretised energy norm. We then establish that schemes in which the water surface elevations are stored at the mesh triangle centroids do not share these advantageous symmetry properties. Finally we show examples which demonstrate that the centroid based scheme is subject to unstable growing modes, particularly in long timescale, Coriolis dominated problems; while the energy conservative circumcentre based scheme suffers from no such limitation. We conclude that unstructured C-grid methods using the triangle circumcentres and the conservative Coriolis scheme derived here therefore have advantages for this sort of problem over those schemes based on centroids.     

A numerical study for some efficient time-integration schemes using barotropic equations

Three kinds of methods, i. e., explicit, semi-implicit, and semi-implicit and semi-Lagrangian method, are tested in the time-integration of shallow-water equations on rotating sphere. Helpful results are available from experiments, especially about the accuracy and efficiency of different semi-implicit and semi-Lagrangian schemes.     

Assimilation of the data of observations and adaptive prediction of the natural processes

We propose a dynamical model for the prediction of random components of natural processes. The model is based on the system concept of adaptive balance of causes (ABC-model) and contains dynamic equations for the coefficients of influence adapted to the correlations existing in the predicted processes. To improve the accuracy of predictions, we consider two possible schemes of assimilation of the data of observations in the equations of the ABC-model, namely, the Kolmogorov and Kalman schemes. Both schemes are oriented toward the application of sample correlation coefficients for the prediction of time series of measurements and, hence, take into account the nonstationarity of actual natural processes. We present some examples of prediction of the simulated time series clarifying the algorithms of assimilation of the data of observations. A conclusion is made that the methods of systems modeling and adaptive prediction of random processes by the ABC-method are quite promising.     

A Vertical Two-Dimensional Model to Simulate Tidal Hydrodynamics in A Branched Estuary

A vertical (laterally averaged) two-dimensional hydrodynamic model is developed for tides, tidal current, and salinity in a branched estuarine system. The goveming equations are solved with the hydrostatic pressure distribution assumption and the Boussinesq approximation. An explicit scheme is employed to solve the continuity equations. The momentum and mass balance equations are solved implicitly in the Cartesian coordinate system. The tributaries are govemed by the same dynamic equations. A control volume at the junctions is designed to conserve mass and volume transport in the finite difference schemes, based on the physical principle of continuum medium of fluid. Predictions by the developed model are compared with the analytic solutions of steady wind-driven circulatory flow and tidal flow. The model results for the velocities and water surface elevations coincide with analytic results. The model is then applied to the Tanshui River estuarine system. Detailed model calibration and verification have been conducted with measured water surface elevations,tidal current, and salinity distributions. The overall performance of the model is in qualitative agreement with the available field data. The calibrated and verified numerical model has been used to quantify the tidal prism and flushing rate in the Tanshui River-Tahan Stream, Hsintien Stream, and Keelung River.     

Implementation of high-order particle-tracking schemes in a water column model

Stochastic differential equations (SDEs) offer an attractively simple solution to complex transport-controlled problems, and have a wide range of physical, chemical, and biological applications, which are dominated by stochastic processes, such as diffusion. As for deterministic ordinary differential equations (ODEs), various numerical scheme exist for solving SDEs. In this paper various particle-tracking schemes are presented and tested for accuracy and efficiency (time vs. accuracy). To test the schemes, the particle tracking algorithms are implemented into a community wide used 1D water column model. Modelling individual particles allows a straightforward physical interpretation of the involved processes. Further, this approach is strictly mass conserving and does not suffer from the numerical diffusion that plagues grid-based methods. Moreover, the Lagrangian framework allows to assign properties to the individual particles, that can vary spatially and temporally. The movement of the particles is described by a stochastic differential equation, which is consistent with the advection–diffusion equation. Here, the concentration profile is represented by a set of independent moving particles, which are advected according to the velocity field, while the diffusive displacements of the particles are sampled from a random distribution, which is related to the eddy diffusivity field.The paper will show that especially the 2nd order schemes are accurate and highly efficient. At the same level of accuracy, the 2nd order scheme can be significantly faster than the 1st order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.     

An efficient shock capturing algorithm to the extended Boussinesq wave equations

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.     

On the divergence-form finite-difference approximation to the momentum advection in curvilinear orthogonal coordinates

The traditional finite-difference schemes for the dynamical equations in curvilinear orthogonal coordinates have a basic flaw: they conserve only energy, but not momentum. A finite-difference approximation to these equations is suggested that conserves both energy and momentum.     

解四阶拟线性波动方程的一类二阶差分格式

建立了解一类四阶拟线性耗散、色散波动方程初边值问题的Crank-Nicolson差分方法，并结合外推的技巧，给出了1个线性化方法；证明了差分解的存在唯一性；用能量估计的方法证明了此格式的二阶收敛性和无条件稳定性；给出了一些数值结果。     

High-resolution central difference scheme for the shallow water equations

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…     

A numerical scheme for morphological bed level calculations

The typical equation for bed level change in sediment transport in river, estuary and near shore systems is based on conservation of sediment mass. It is generally a nonlinear conservation equation for bed level. The physics here are similar to shallow water wave equations and gas dynamics equation which will develop shock waves in many circumstances. Many state-of-art morphological models use classical lower order Lax–Wendroff or modified Lax–Wendroff schemes for morphology which are not very stable for long time sediment transport processes simulation. Filtering or artificial diffusion are often added to achieve stability. In this paper, several shock capturing schemes are discussed for simulating bed level change with different accuracy and stability behaviors. The conclusion is in favor of a fifth order Euler-WENO scheme which is introduced to sediment transport simulations here over other schemes. The Euler-WENO scheme is shown to have significant advantages over schemes with artificial viscosity and filtering processes, hence is highly recommended especially for phase-resolving sediment transport models.     

A Terrain-Following Crystal Grid Finite Volume Ocean Circulation Model

A three dimensional hydrostatic finite volume ocean model has been developed to solve the integral dynamical equations. Since the basic (integral) equations are solved for finite volumes rather than grid points, the flux conservation is easily enforced, even on arbitrary meshes. Both upwind and high-order combined compact schemes can be incorporated into the model to increase computational stability and accuracy. This model uses a highly distorted grid system near the boundary. The lateral boundaries of each finite volume are perpendicular to x and y axes and the two vertical boundaries are not purely horizontal. Four types of finite volumes are designed to follow the terrain with four (Type-A), three (Type-B), two (Type-C), and one (Type-D) vertices in the lower surface. Such a terrain-following grid discretization has superior features to z- and σ-coordinate systems. The accuracy of this model was tested.     

Dispersion and stability analyses of the linearized two-dimensional shallow water equations in Cartesian coordinates

In the present study, a Fourier analysis is used to develop expressions for phase and group speeds for both continuous and discretized, linearized two-dimensional shallow water equations, in Cartesian coordinates. The phase and group speeds of the discrete equations, discretized using a three-point scheme of second order, five-point scheme of fourth order and a three-point compact scheme of fourth order in an Arakawa C grid, are calculated and compared with the corresponding values obtained for the continuous system. The three-point second-order scheme is found to be non-dispersive with grid resolutions greater than 30 grids per wavelength, while both the fourth-order schemes are non-dispersive with grid resolutions greater than six grids per wavelength. A von Neumann stability analysis of the two- and three-time-level temporal schemes showed that both schemes are stable. A wave deformation analysis of the two-time-level Crank–Nicolson scheme for one-dimensional and two-dimensional systems of shallow water equations shows that the scheme is non- dispersive, independent of the Courant number and grid resolution used. The phase error or the dispersion of the scheme decreases with a decrease in the time step or an increase in grid resolution.     

Run-up heights of nearshore tsunamis based on quadtree grid system

To investigate the run-up heights of nearshore tsunamis in the vicinity of a circular island, a numerical model has been developed based on quadtree grids. The governing equations of the model are the nonlinear shallow-water equations. The governing equations are discretized explicitly by using the leap-frog scheme on adaptive hierarchical quadtree grids. The refined quadtree grids are generated around a circular island on a combined domain of rectangular and circular grids. The predicted numerical results have been verified by comparing to available laboratory measurements. A good agreement has been observed.     

An implicit method using contravariant velocity components and its application to calculations in a harbour-channel area

lNTRODUCTIONAsoneofthenumericalcalculationmethodsinvolvingfluiddynamicsinnearshoreareas,theboundaryfittedgridmethodhasmanyadvantagessuchasfittingboundaries,beingsuitableforengineeringconstructionswithsmallscalesandimprovingtheaccuracybydensifyinggridpointsintheinterestedareas'Incomparisonwiththefiniteelementmethodwithnon-uniformgrids,theboundary-fittedgridmodelismorewidelyusedbecauseofitssuperioritiesinusingthematurefi-nitedifferenceschemeandinoccupyingsma1lcomputermemory(Sheng,l986;Haase.…     

流速逆变张量隐式求解方法及其在航道港池流场计算中的应用

运用高分辨率的边界适应网格进行流体动力学数值计算时,如何提高计算稳定性和减少计算量成为数值求解的关键性问题.在非正交的边界适应坐标系中,每个动量方程中同时出现了两个交叉方向的水位偏导数项,给隐式求解带来困难,而显式格式下的时间步长由于受与空间步长有关的Courant-Friedrichs-Lewy条件限制,计算量成倍增加.本文从广义曲线坐标系下浅海动力学方程组出发,导出了流速的逆变张量所满足的动量方程组,使方程中的水位偏导数项变成了沿某一协变基向量方向占优的形式,方便地采用了交替方向隐式差分格式,从而提高了计算稳定性并减小了计算量.本文通过对澳门海域航道和港池中流场的计算,证实了该模式是一种进行高分辩率数值计算的有效方法.     

Calculations of the evolution of the upper ocean based on the similarity theory

Several schemes of turbulent mixing in the upper ocean are considered, including a modified scheme based on the modified Monin-Obukhov similarity theory. The schemes have been used for the calculation of the evolution of the upper ocean. The results are compared with the data of automated buoys. It is shown that the scheme based on the similarity theory gives a result not worse than the commonly used ones and has several advantages, which makes it the most appropriate for including in the ocean circulation models and climate models.     

An Explicit High Resolution Scheme for Nonlinear Shallow Water Equations

The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe＇ s flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws （MUSCL） variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.