Numerical multi-block grids in coastal ocean circulation modeling

In coastal ocean modeling, one desires to capture the evolution and interaction of multi-scales of physical phenomena in a complicated physical domain. With limited computer resources, an appropriate choice of the numerical grid has a key role in determining the quality of the solution of a numerical coastal ocean model. Traditionally, single-block rectangular grids have been most commonly used in coastal ocean modeling for their simplicity. An effective coastal ocean model represents the dynamics of the coastal ocean flow on a numerical grid, including the effects of complicated features such as coastlines, bottom topography (submarine canyons, seamounts, narrow straits), and multi-scale physical phenomena. These problems require a model grid system more efficient than a traditional single-block rectangular grid. The model grids must give better resolution of coastlines and boundary conditions, multi-scale physical phenomenon, and save computer resources. These grids can also easily increase horizontal resolution in a subregion of the model domain without increasing computer expense with high resolution over the entire domain. The multi-block numerical generation grid technique is used in developing a coastal ocean system applied to the Mediterranean Sea (MED) with complicated coastlines, bottom topography and multi-scale physical features. The MED coastal ocean system consists of the MED model based on the Princeton Ocean Model, numerical grid generation routines, and a grid package which allows the model to be coupled with model grids. The traditional, nine-block orthogonal grid, and eight-block curvilinear nearly orthogonal coastline-following grid are used in the study. The numerical solutions with the three grids are compared in term of effectiveness. The numerical simulations show some MED basic physical features.     

Finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains

A finite difference time-dependent numerical method for the wave equation, supported by recently derived novel elliptic grids, is analyzed. The method is successfully applied to single and multiple two-dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries. The new grids have nearly uniform cell area (J-grids) and nearly uniform grid line spacing (αγ-grids). Numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state. The restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing. As a consequence, moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost. Also, numerical solutions for wave problems inside annular regions of complex shapes are obtained. The use of the new grids results in late time stability in contrast with other classical finite difference time-dependent methods.     

The influence of shelf zone topography and coastline geometry on coastal trapped waves

The results of numerical experiments with a model of coastal trapped waves are presented to identify two important features for regional modeling of the interaction of a shelf zone with open ocean. First, a wave train of this type can be formed by wind action at a considerable distance from the place of impact. The waves propagate along a coastline without significant loss of energy, provided that the coastline and shelf zone topography have no features comparable to the Rossby radius. However, the waves lose energy while passing over capes and submarine canyons and when the shelf width decreases. For regional modeling, remote generation of waves must be thoroughly investigated and taken into account. The other feature is that the propagating waves can use part of energy to form density anomalies on the shelf by raising intermediate waters from the adjacent offshore areas of the open ocean. Thus, coastal trapped waves can carry wind energy from wind action areas to other coastal areas to form density anomalies and other types of motion.     

A new type of boundary treatment for wave propagation

Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example. A serious problem with uniform grids is the approximation of boundary conditions at a boundary not aligned with the grid. In this paper, boundary conditions are introduced by modifying appropriate material coefficients at a few grid points close to the embedded boundary. This procedure is applied to the Yee scheme and the resulting method is proven to be \(L^2\)-stable in one space dimension. Depending on the boundary approximation technique it is of first or second order accuracy even if the boundary is located at an arbitrary point relative to the grid. This boundary treatment is applied also to a higher order discretization resulting in a third order accurate method. All algorithms have the same staggered grid structure in the interior as well as across the boundaries for efficiency. A numerical example with the extension to two space dimensions is included.     

Application of the spherical metric tensor to grid adaptation and the solution of applied problems

New results concerning the construction and application of adaptive numerical grids for solving applied problems are presented. The grid generation technique is based on the numerical solution of inverted Beltrami and diffusion equations for a monitor metric. The capabilities of the spherical metric tensor as applied to adaptive grid generation are examined in detail. Adaptive hexahedral grids are used to numerically solve a boundary value problem for the three-dimensional heat equation with a moving boundary in a continuous medium with discontinuous thermophysical parameters; this problem models the interaction of a thermal wave with a thermocouple embedded in the solid.     

Generation of 3D grids by approximate factorization

An efficient procedure is proposed for generating three-dimensional numerical grids inside domains with curvilinear boundaries. Systems of partial differential equations of elliptical type are used as the basic equations for grid generation. An approximate factorization method is proposed for numerical solution of the resulting boundary-value problem. Calculation results are reported for a prototype domain. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 99–105, 1999.     

A simple and effective mesh adaptation

Simulation accuracy is greatly influenced by grid quality. Here, mesh quality indicates orthogonality of grid lines at the boundaries and quasi-orthogonality within critical regions, smoothness, solution adaptive behaviour and bounded aspect ratios. A simple, effective and computationally efficient approach for adapting quadrilateral grids to a given adaptive functional is presented. Several numerical examples are explored for supporting our claim.     

Solving the problem of non-stationary filtration of substance by the discontinuous Galerkin method on unstructured grids

A numerical algorithm is proposed for solving the problem of non-stationary filtration of substance in anisotropic media by the Galerkin method with discontinuous basis functions on unstructured triangular grids. A characteristic feature of this method is that the flux variables are considered on the dual grid. The dual grid comprises median control volumes around the nodes of the original triangular grid. The flux values of the quantities on the boundary of an element are calculated with the help of stabilizing additions. For averaging the permeability tensor over the cells of the dual grid, the method of support operators is applied. The method is studied on the example of a two-dimensional boundary value problem. The convergence and approximation of the numerical method are analyzed, and results of mathematical modeling are presented. The numerical results demonstrate the applicability of this approach for solving problems of non-stationary filtration of substance in anisotropic media by the discontinuous Galerkin method on unstructured triangular grids.     

The effects of interplay between the rotation and shoaling for a solitary wave on variable topography

This paper presents specific features of solitary wave dynamics within the framework of the Ostrovsky equation with variable coefficients in relation to surface and internal waves in a rotating ocean with a variable bottom topography. For solitary waves moving toward the beach, the terminal decay caused by the rotation effect can be suppressed by the shoaling effect. Two basic examples of a bottom profile are analyzed in detail and supported by direct numerical modeling. One of them is a constant‐slope bottom and the other is a specific bottom profile providing a constant amplitude solitary wave. Estimates with real oceanic parameters show that the predicted effects of stable soliton dynamics in a coastal zone can occur, in particular, for internal waves.     

地形构造中地震波传播的非对称交错网格模拟

提出了一种新的三维空间对称交错网格差分方法,模拟地形构造中弹性波传播过程．通过具有二阶时间精度和四阶空间精度的不规则网格差分算子用来近似一阶弹性波动方程,引入附加差分公式解决非均匀交错网格的不对称问题．该方法无需在精细网格和粗糙网格间进行插值,所有网格点上的计算在同一次空间迭代中完成．使用精细不规则网格处理海底粗糙界面、 断层和空间界面等复杂几何构造, 理论分析和数值算例表明, 该方法不但节省了大量内存和计算时间, 而且具有令人满意的稳定性和精度．在模拟地形构造中地震波传播时,该方法比常规方法效率更高．     

Improving multi-block sigma-coordinate for 3D simulation of sediment transport and steep slope bed evolution

This paper aims at developing a multi-block sigma-coordinate to simulate morphological evolutions. The developed multi-block sigma-coordinate can represent a steep slope topography smoothly with different numbers of horizontal layers, without producing any truncation error and artificial flux (PGFE). The multi-block sigma–coordinate can easily increase the depth-direction resolution of the sediment transport module in the sub-regions, without the aggregation of computational points in the shallow areas which may be caused by using high resolution over the entire domain. The model is beneficial for long-term simulation of morphological evaluations in lakes where the bed slope near delta region (the sedimentary area which forms where a river enters a lake/ocean) is mostly steep, and delta keeps advancing down the lake.The multi-block sigma approach at the block interface, where the number of horizontal layers varies, allocates the flux to the neighboring cells according to two essential factors, namely “the common border length” and “satisfying continuity” which lead to defining the virtual cells. A series of numerical tests have been performed. Comparison between the numerical results, the analytical solutions, and the experimental data demonstrated an appreciable accuracy, a satisfactory performance and the efficiency of this scheme.     

A New Algorithm for Generating Tetrahedral Grids

A new algorithm is presented for generating tetrahedral grids for bound domains of complicated structure and form that consist of the union of several subdomains. The grids obtained using the algorithm have regularity, in that there is no tetrahedron whose vertices lie at different sides of the subdomain boundaries. The algorithm has high fast-action: the time needed to generate a grid consisting of 10⁵?10⁶ tetrahedrons is approximately 20–40 seconds of the work of a standard personal computer. All steps of the algorithm are described in detail, and examples of generated grids are given.     

Generation and dissipation of ocean surface waves

Ocean surface waves constitute one of the most important sources of external forces that act on ships and offshore structures. Most ocean waves are generated by wind, but various other effects such as currents, ground and coastal topology, breaking and wave–wave interaction have an influence on the growth and dissipation of wave energy at specific frequency ranges. These water waves are inherently random in nature and their exact shape is difficult to describe, even when confining the described area to a small range. While many different approaches exist to describe the spectral characteristics of ocean waves, some of the processes which affect the generation of waves are still poorly understood. This paper addresses some techniques which serve to describe seaway spectra with respect to the subsequent analysis of dynamic mechanical systems in the ocean such as ships, platforms and pipelines. Advantages and limits of the different approaches are discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Breaking of two-dimensional relativistic electron oscillations under small deviations from axial symmetry

A numerical asymptotic model for the breaking of two-dimensional plane relativistic electron oscillations under a small deviation from axial symmetry is developed. The asymptotic theory makes use of the construction of time-uniformly applicable solutions to weakly nonlinear equations. A special finite-difference algorithm on staggered grids is used for numerical simulation. The numerical solutions of axially symmetric one-dimensional relativistic problems yield two-sided estimates for the breaking time. Some of the computations were performed on the “Chebyshev” supercomputer (Moscow State University).     

Generation of structured difference grids in two-dimensional nonconvex domains using mappings

Generation of structured difference grids in two-dimensional nonconvex domains is considered using a mapping of a parametric domain with a given nondegenerate grid onto a physical domain. For that purpose, a harmonic mapping is first used, which is a diffeomorphism under certain conditions due to Rado’s theorem. Although the harmonic mapping is a diffeomorphism, its discrete implementation can produce degenerate grids in nonconvex domains with highly curved boundaries. It is shown that the degeneration occurs due to approximation errors. To control the coordinate lines of the grid, an additional mapping is used and universal elliptic differential equations are solved. This makes it possible to generate a nondegenerate grid with cells of a prescribed shape.     

Generating Boundary Conditions for the Calculation of Tsunami Propagation on Nested Grids

Some boundary conditions used to numerically simulate tsunami generation and propagation are studied. Special attention is given to generating boundary conditions thatmake it possible to simulate tsunami waves with desired characteristics (amplitude, time period and, in general, waveform). Since the water flow velocity in a propagating tsunami wave is uniquely defined by its height and ocean depth, one can simulate a wave propagating from the boundary into the simulation area. This can be done by specifying the wave height and water flow velocity on the boundary. This method is used to numerically simulate the propagation of a tsunami from the source to the coast on a sequence of refined grids. In this numerical experiment the wave parameters are transferred from the larger area to the subarea via boundary conditions. This method can also generate a wave that has certain characteristics on a specified line.     

Numerical model for the shallow water equations on a curvilinear grid with the preservation of the Bernoulli integral

Methodological aspects concerning the construction of a two-dimensional numerical model for reservoir flows based on the shallow water equations are considered. A numerical scheme is constructed by applying the control volume method on staggered grids in combination with the Bernoulli integral, which is used to interpolate the desired fields inside a grid cell. The implementation of the method yields a monotone numerical scheme. The results of numerical integration are compared with the exact solution.     

Numerical simulation of seismic activity by the grid-characteristic method

Seismic activity in homogeneous and layered enclosing rock masses is studied. A numerical mechanical-mathematical model of a hypocenter is proposed that describes the whole range of elastic perturbations propagating from the hypocenter. Synthetic beachball plots computed for various fault plane orientations are compared with the analytical solution in the case of homogeneous rock. A detailed analysis of wave patterns and synthetic seismograms is performed to compare seismic activities in homogeneous and layered enclosing rock masses. The influence exerted by individual components of a seismic perturbation on the stability of quarry walls is analyzed. The grid-characteristic method is used on three-dimensional parallelepipedal and curvilinear structured grids with boundary conditions set on the boundaries of the integration domain and with well-defined contact conditions specified in explicit form.     

On the Calculation of Border and Contact Nodes by a Grid-Characteristic Method on Non-Periodic Tetrahedral Grids

A grid-characteristic method for the numerical simulation of wave processes in continuum mechanics was initially proposed, and has been successfully applied to periodic hexagonal computational grids. Later it was proposed to adapt this method to non-periodic triangle and tetrahedral grids, and wide computational experience has been gained. However, this approach encounters some difficulties in the calculation of border and contact points when applied to various grid configurations in areas with complex geometries. In this paper, limitations of the method which cause such problems are considered, and some improvements to overcome them are proposed.