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The vadose zone is the portion of the subsurface above the water table and its pore space usually contains air and water. Due to the presence of infiltration, erosion, plant growth, microbiota, contaminant transport, aquifer recharge, and discharge to surface water, it is crucial to predict the transport rate of water and other substances within this zone. However, flow in the vadose zone has many complications as the parameters that control it are extremely sensitive to the saturation of the media, leading to a nonlinear problem. This flow is referred as unsaturated flow and is governed by Richards equation. Analytical solutions for this equation exists only for simplified cases, so most practical situations require a numerical solution. Nevertheless, the nonlinear nature of Richards equation introduces challenges that causes numerical solutions for this problem to be computationally expensive and, in some cases, unreliable. High order mimetic finite difference operators are discrete analogs of the continuous differential operators and have been extensively used in the fields of fluid and solid mechanics. In this work, we present a numerical approach involving high order mimetic operators along with a Newton root-finding algorithm for the treatment of the nonlinear component. Fully-implicit time discretization scheme is used to deal with the problem’s stiffness.

While global- and basin-scale processes can be captured quite well with computationally-inexpensive hydrostatic models, smaller-scale features such as shoaling nonlinear internal waves and bores, coastal fronts, and other convective processes require the use of a nonhydrostatic model to accurately capture dynamics. Here the nonhydrostatic capabilities of the General Curvilinear Coastal Ocean Model (GCCOM) in a stratified environment are introduced. GCCOM is a three-dimensional, nonhydrostatic Large Eddy Simulation (LES), rigid lid model that has the ability to run in a fully three-dimensional general curvilinear coordinate system. This model was previously validated for unstratified flows with curvilinear coordinates. Here, recent advances of the model to simulate stratified flows are presented, focusing on sigma coordinate grids with both flat bottom geometry and a local gently sloping seamount. In particular, a suite of test cases widely used as benchmarks for assessing the nonhydrostatic capabilities for gravity-driven flows and internal waves is presented: an internal seiche in a flat bottom tank, the classic lock release and gravity current experiment, and a field-scale internal wave beam experiment consisting of an oscillating tidal flow over a topographic ridge. GCCOM shows excellent agreement with the benchmark test cases and is able to accurately resolve complex nonhydrostatic phenomena in stratified flows. Future studies will utilize the model capabilities for realistic field-scale internal wave simulations.

Large scale geophysical modeling uses high performance computing systems to expedite the solutions of very large, complex systems. High disk latencies, low IOPS, and low read/write data transfer rates are relegating many numerical simulations to I/O bound jobs, where the run time is bound not by CPU rate, but by I/O rate. In this paper we seek to improve the I/O of two geophysical modeling applications and take full advantage of the parallel nature of the programs, as well as the file management system for the large output files. Parallelizing output for these programs is achieved using PnetCDF, a parallel implementation of the netCDF format, and BeeGFS, an open-source parallel file system. Using these solutions, we have significantly decreased the amount of time spent saving data to disk, and give analysis of the features used in relation to PnetCDF with BeeGFS I/O optimization.

This paper introduces a new software package, written in MATLAB ® , that generates an extended discrete Laplacian ( L = D G = ∇ ⋅ ∇ ) based on the Castillo–Grone Mimetic difference operators over a general curvilinear grid. The boundary conditions are included in the extended discrete Laplacian Operator, i.e. Dirichlet, Neumann, as well as Robin boundary conditions. The user only needs to provide a curvilinear grid, a desired operator order, and the degree of the interpolating polynomial. The operator is tested in different condition and its performance is reported. Finally, based on accuracy of the results, amount of required memory, and the computation that is needed, it is concluded that the 4th order operator is the best one.

This paper describes how discrete versions of the derivative on the real line induce discrete version of the gradient and divergence in higher dimensions. This is geometrically motivated by results in algebraic graph theory since the grid in n dimensions is the graph Kronecker product of the path on n vertices. The resulting technique relies heavily on the matrix Kronecker product, and is an analogue of a derivation in multilinear algebra.

The oceans effect is predominant of Earth systems. From weather and food supplies to shipping routes, the oceans impact everyone, which makes the numerical modeling of ocean phenomenon incredibly important. With as many different ways the ocean impacts the Earth, there are just as many models describing its behavior. Since there are such different physical parameters in each problem, and with many different techniques available, no single ocean model can do everything in a computationally efficient way. Deep water models may not able to capture surface conditions or violent interactions accurately, while shallow water models usually lack the sophistication to render slow processes impacted by temperature, pressure, and salinity. Coupling is often used to combine the best features of different models to better resolve complex interactions, that each model alone would be unable to solve fully. Presented here is a coupling between a non-hydrostatic, rigid lid model, the General Curvilinear Coastal Ocean Model (GCCOM) a hydrostatic free surface model, the Shallow Water At Shore model (SWASH), interfacing by using the Distributed Coupling Toolkit (DCT). To test the coupling, a lock release experiment is modeled, to show the detail of internal processes and the surface features from internal waves striking the surface. The coupled system is shown to be faster and scales better than the SWASH model alone running in its non-hydrostatic capacity.

The General Curvilinear Coastal Ocean Model (GCCOM) is a 3D curvilinear, structured-mesh, non-hydrostatic, large-eddy simulation model that is capable of running oceanic simulations. GCCOM is an inherently computationally expensive model: it uses an elliptic solver for the dynamic pressure, meter-scale simulations requiring memory footprints on the order of 10 12 cells and terabytes of output data. As a solution for parallel optimization, the Fortran-interfaced Portable&ndash, Extensible Toolkit for Scientific Computation (PETSc) library was chosen as a framework to help reduce the complexity of managing the 3D geometry, to improve parallel algorithm design, and to provide a parallelized linear system solver and preconditioner. GCCOM discretizations are based on an Arakawa-C staggered grid, and PETSc DMDA (Data Management for Distributed Arrays) objects were used to provide communication and domain ownership management of the resultant multi-dimensional arrays, while the fully curvilinear Laplacian system for pressure is solved by the PETSc linear solver routines. In this paper, the framework design and architecture are described in detail, and results are presented that demonstrate the multiscale capabilities of the model and the parallel framework to 240 cores over domains of order 10 7 total cells per variable, and the correctness and performance of the multiphysics aspects of the model for a baseline experiment stratified seamount.

Fractured geologic media can yield anisotropies in solute and heat diffusion due to the formation of changing fluid network connectivity in a rock matrix. In this paper we model Steady-state anisotropic heat diffusion as an elliptic partial differential equation with a symmetric positive definite second rank thermal conductivity tensor. We model diffusive flux as a non-diagonal symmetric tensor, which can potentially have jump discontinuities that are not aligned with the coordinate axis. The presence of jump discontinuities due to joints and faults in a rock matrix impose difficulties on existing, well-established numerical schemes that model diffusive transport. In our scheme, we model diffusive flux using mimetic finite difference operators, which are discrete analogs of the classical continuous differential operators. We introduce a 2 nd - and 4 th -order mimetic formulation for computing anisotropic fluxes. Numerical results demonstrate our formulation yields a substantial improvement compared to similar mimetic schemes.

We present high-order mimetic finite-difference operators that satisfy the extended Gauss theorem. These operators have the same order of accuracy in the interior and at the boundary, no free parameters and optimal bandwidth. They are defined over staggered grids, using weighted inner products with a diagonal norm. We present several examples to demonstrate that mimetic finite-difference schemes using these operators produce excellent results.

High order mimetic finite difference operators that satisfy a discrete extended Gauss Divergence theorem are presented. These operators have the same order of accuracy in the interior as well as the boundary, no free parameters and optimal bandwidth. They are constructed on staggered grids, using weighted inner products with a diagonal norm. We present several examples to demonstrate that mimetic finite difference schemes using these operators produce excellent results.

A 3D hydrodynamic model (Delf3D) was applied for San Quintin Bay (SQB), B.C., Mexico. Calibration and validation were conducted using measured bathymetry, wind, water surface elevation and velocities fields. The calibration period took place during the winter of 2010. Model predictions were evaluated graphically and statistically against field observations in order to quantify the accuracy of the predictions and evaluate the success of the model calibration. A 2D Calibration of tide constituents at the boundaries, seabed roughness and depths was done to evaluate the improvement in the velocities for the 3D Model. Comparisons between the model predictions and the field observations of the water surface elevation at interior stations indicated that the model had been successfully calibrated. It also indicated that the model predictions were highly correlated with the observed water surface elevations and velocities. This connection between the observed and simulated values was detected through graphical comparisons and root-mean-square errors and correlation coefficient. The objective of this study was to demonstrate that OpenDA could be used effectively to rapidly calibrate a hydrodynamic model for smaller sub-meso scales like the size of SQ Bay.

We focus on the construction of 2- and 3D mimetic gradient, divergence, curl, and Laplacian operators. We base this work on the method by Castillo and Grone, which constructs mimetic gradient and divergence operators via a discrete instance of Gauss’ divergence theorem. This method can not construct tenth-order gradient nor eighth-order divergence operators (nor higher) because the computed weights discretizing the corresponding weighted inner products are not all positive for these cases. Thus, we define the tenth order and the eighth order thresholds as critical orders of accuracy for the gradient and divergence operators, respectively. In previous works, we introduced the Castillo–Blomgren–Sanchez algorithm. This algorithm constructs supercritical-order mimetic operators. The contribution of this work is the extension to higher dimensions of the operators constructed by this algorithm. This includes detailing the mathematics of this extension. We also detail the construction of a mimetic curl operator via a linear combination of the divergence of auxiliary Gaussian fluxes. This avoids any interpolation from classic discretization approaches based on Stokes’ theorem. We validate our operators by solving higher-dimensional elliptic problems.

Mimetic difference operators Div, Grad and Curl, have been constructed to provide a high order of accuracy in numerical schemes that mimic the properties of their corresponding continuum operators; hence they would be faithful to the physics. However, this faithfulness of the discrete basic operators might not be sufficient if the numerical difference scheme introduces some numerical energy increase, which would obviously result in a potentially unstable performance. We present a high order compact mimetic scheme for 2D wave motions and show that the energy of the system is also conserved in the discrete sense.

We introduce the Mimetic Methods Toolkit (MTK), an object-oriented Application Programming Interface, implementing Mimetic Finite Differences to assist in the development of scientific applications where the numerical solution of Partial Differential Equations is required. The MTK’s design is based on a variant of the Castillo–Grone Method for the construction of discrete differential operators that mimic important properties of their continuous counterparts. The MTK is built as a collection of abstract and concrete classes, thus allowing for an extensible framework, which fosters code reutilization, while intuitively educating the user about the theoretical aspects of Mimetic Finite Differences. We present an overview to Mimetic Finite Differences, and we discuss the computational modeling of the related concepts; in this way, we explain how the MTK implements these methods. In this article, we present examples to illustrate the MTK’s usage philosophy and the validity of the implementation of the MTK by comparing our results with previously studied reference solutions.

Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

In elastic media, finite-difference (FD) implementations of free-surface (FS) boundary conditions on partly staggered grid (PSG) use the highly dispersive vacuum formulation (VPSG). The FS boundary is embedded into a “vacuum” grid layer (null Lame’s constants and negligible density values) where the discretized equations of motion allow computing surface displacements. We place a new set of compound (stress-displacement) nodes along a planar FS and use unilateral mimetic FD discretization of the zero-traction conditions for displacement computation (MPSG). At interior nodes, MPSG reduces to standard VPSG methods and applies fourth-order centered FD along cell diagonals for staggered differentiation combined with nodal second-order FD in time. We perform a dispersion analysis of these methods on a Lamb’s problem and estimate dispersion curves from the phase difference of windowed numerical Rayleigh pulses at two FS receivers. For a given grid sampling criterion (e.g., six or ten nodes per reference S wavelength λS), MPSG dispersion errors are only a quarter of the VPSG method. We also quantify root-mean-square (RMS) misfits of numerical time series relative to analytical waveforms. MPSG RMS misfits barely exceed 10 % when nine nodes sample the minimum S wavelength \(\lambda _{\text {MIN}}^{\mathrm {S}}\) in transit (along distances \(\sim \)145\(\lambda _{\text {MIN}}^{\mathrm {S}}\)). In same tests, VPSG RMS misfits exceed 70 %. We additionally compare MPSG to a consistently fourth-order mimetic method designed on a standard staggered grid. The latter equates the former’s dispersion errors on grids twice denser and shows higher RMS precision only on grids with six or less nodes per \(\lambda _{\text {MIN}}^{\mathrm {S}}\).

In this paper, the Unified Curvilinear Ocean Atmosphere Model (UCOAM) is described. UCOAM is a non-hydrostatic LES model, capable of reading curvilinear orthogonal and non-orthogonal grid in all the three dimensions including the vertical. Furthermore, the interaction of bottom bathymetry with the flow field and particularly, the vertical component of the velocity in a fully curvilinear grid and a sigma grid are presented.

We consider constrained bi-objective optimization problems. One of the extant issues in this area is that of uniform sampling of the Pareto front. We utilize equispacing constraints on the vector of objective values, as discussed in a previous paper dealing with the unconstrained problem. We present a formulation and a dual formulation based on arc-length homotopy continuation and illustrate this method (using standard nonlinear programming tools) on some problems from the literature. We contrast the performance of our method with the results of three other algorithms, showing several orders of magnitude speed-up with respect to evolutionary algorithms, while simultaneously providing perfectly sampled fronts by construction. We then consider a large-scale application: the variational approach to mesh generation for partial differential equations in complex domains. Balancing multiple criteria leads to significantly improved mesh design.

The Regional Ocean Modeling System (ROMS) is a hydrostatic free-surface ocean model ideally suited to simulate mesoscale to basin-scale (10 km – 10000 km) ocean processes. The General Curvilinear Coastal Ocean Model (GCCOM) is a nonhydrostatic large eddy simulation (LES) model designed specifically for high-resolution (meters) simulations. In this research, a hybrid model is developed that nests a fine-grid GCCOM model within a coarse-grid ROMS. The nested GCCOM-ROMS model is tested in an idealized flow over a seamount.

This paper presents the integration of a data assimilation framework and a nonhydrostatic coastal ocean model for the study of turbulent processes and state variables. Interfacing our General Curvilinear Coastal Ocean Model (GCCOM) with NCAR's Data Assimilation Research Testbed (DART), enable the integration of very high-resolution observations into the system. These results included observation system simulation experiments (OSSEs) for test cases using a very steep seamount, by using different observation error variances. Our results demonstrated that the DART-GCCOM model can assimilate high-resolution observations using as few as 30 ensemble members.

While manufacturers are already producing bifacial modules, there is slow adoption into the market place partially due to a current deficiency in bifacial performance and safety standardized testing (for example IEC/UL) which makes comparisons between manufacturers difficult, and in the modeling techniques used to predict the field performance of bifacial systems. A recently developed best-fit model that links effects of the module's minimum height ratio, tilt angle, and ground albedo to the annual energy output is presented in this document. Additionally, we expand on the previously presented bifacial energy model by introducing an azimuthal correction factor for the bifacial energy model, expanding its application space to off-south facing installations.

In this work, we implement and compare two fourth-order compact finite difference (CFD) methods to model the propagation of acoustic waves on a 1-D domain. The first scheme is based on standard implicit CFD constructed on nodal grids and must solve a couple of tridiagonal linear systems at each time iteration. The formulation of the second method uses the novel staggered mimetic CFD that explicitly approximate derivatives at cell center points. Mimetic FD operators inherit this name from the fact that some conservation properties fulfilled by the continuous divergence, gradient, and curl operators, are also satisfied by the discrete mimetic counterparts. Both CFD methods are combined to high-order Runge-Kutta schemes for time integration. In our current preliminary experiments, results show that the mimetic CFD scheme is slightly more accurate and shows similar fourth-order convergence than the nodal scheme. In this paper, we also compare the simulation times spent by the compact schemes to an explicit Leap-frog staggered solver. This comparison reflects the high computational efficiency of the mimetic CFD scheme, and along with its accurate results, suggests the potential of this method for multidimensional wave propagation problems.

We present two fourth-order compact finite difference (CFD) discretizations of the velocity-pressure formulation of the acoustic wave equation in 2-D rectangular grids. The first method uses standard implicit CFD on nodal meshes and requires solving tridiagonal linear systems along each grid line, while the second scheme employs a novel set of mimetic CFD operators for explicit differentiation on staggered grids. Both schemes share a Crank-Nicolson time integration decoupled by the Peaceman-Rachford splitting technique to update discrete fields by alternating the coordinate direction of CFD differentiation (ADI-like iterations). For comparison purposes, we also implement a spatially fourth-order FD scheme using non compact staggered mimetic operators in combination to second-order leap-frog time discretization. We apply these three schemes to model acoustic motion under homogeneous boundary conditions and compare their experimental convergence and execution times, as grid is successively refined. Both CFD schemes show four-order convergence, with a slight superiority of the mimetic version, that leads to more accurate results on fine grids. Conversely, the mimetic leap-frog method only achieves quadratic convergence and shows similar accuracy to CFD results exclusively on coarse grids. We finally observe that computation times of nodal CFD simulations are between four and five times higher than those spent by the mimetic CFD scheme with similar grid size. This significant performance difference is attributed to solving those embedded linear systems inherent to implicit CFD. A new application of mimetic fourth-order finite differences coupled to leap-frog time integration to wave propagation phenomena.A new 2-D compact formulation on nodal meshes with Crank-Nicolson time integration.A new mimetic formulation based on leap-frog scheme and Crank-Nicolson time integration on a 2-D staggered grids.Two new fourth-order compact finite difference schemes to model acoustic motion under homogeneous boundary conditions.

SUMMARY We incorporate rate- and state-dependent friction in explicit finite difference (FD) simulations of mode II dynamic ruptures in elastic media, using the Mimetic Operators Split-Node (MOSN) method, with adjustable order of spatial accuracy (second-, fourth- or mixed-order accurate), including an option that is fourth-order accurate at the fault discontinuity as well as in the elastic volume. At fault points, the rate and state equations combined with the spatially discretized momentum conservation equations form a coupled system of ordinary differential equations (ODEs) for slip velocity and state variable. As a consequence of the rapid damping of velocity perturbations due to the direct effect, this system exhibits numerical stiffness that is inversely proportional to velocity squared. Approximate solutions to this velocity-state system are achieved by two different implicit schemes: (i) a fourth-order Rosenbrock integration of the full system using multiple substeps and (ii) low order integrations (backward Euler and trapezoidal) of the velocity equation, time-staggered with analytic integration of the state equation under the approximation of constant slip velocity over the time step. In assessing the numerical schemes, we use three test problems: ruptures with frictional resistance controlled by (i) a slip evolution law with strong velocity-weakening behaviour at high slip rates, representing thermal weakening due to flash heating of microscopic asperity contacts, (ii) the classic (low-velocity) slip evolution law and (iii) the classic aging evolution law. A convergence analysis is carried out using reference solutions from a spectral boundary integral equation method (BIEM) (a method restricted to homogeneous media, with nominal spectral accuracy in space and second-order accuracy in time for smooth solutions). Errors are measured by root-mean-square differences of fault-plane time histories (slip, slip rate, traction and state). MOSN shows essentially the same convergence rates as BIEM: second-order convergence for slip and state-variable misfits, with slower (but at least first-order) convergence for slip rates and tractions. For a given grid spacing, fourth-order MOSN is as accurate as BIEM for all variables except slip-rate. MOSN-Rosenbrock nominally has fourth-order temporal accuracy for the fault-plane velocity-state ODE integration (compared to lower-order accuracy for the other two MOSN schemes) and therefore provides an important theoretical benchmark. However, it is sensitive to details of the elastic calculation scheme and occasionally its adaptive substepping performs poorly, leading to large excursions from the reference solution. In contrast, MOSN-trapezoidal is robust and reliable, much easier to implement than MOSN-Rosenbrock, and in all cases achieves precision as good as the latter without recourse to substepping. MOSN-Euler has the same advantages as MOSN-trapezoidal, except that its nominal first-order temporal accuracy ultimately leads to larger errors in slip and state variable compared with the higher-order MOSN schemes at sufficiently small grid spacings and time steps.

SUMMARY We present a split-node finite difference (FD) method for modelling shear ruptures that is consistently fourth-order accurate in its spatial discretization, both in the interior of the model and at the fault. The method, called mimetic operator split node (MOSN), uses a staggered grid, and the fault plane is discretized using split nodes for both discontinuous displacement and discontinuous stress components. The method differs in several ways from previous FD methods for rupture modelling. (1) It uses one-sided differentiation (in the fault-normal direction) to retain high-order accuracy at the discontinuity. (2) The one-sided differentiation is implemented with mimetic FD operators that conserve the integration-by-parts formula (including boundary terms) in discrete form. (3) The mimetic operators lead naturally to the evaluation of conjugate tractions at the split-node displacement sites, as required to apply the frictionalsliding boundary conditions. In the interior of the grid, the mimetic operators reduce to the conventional fourth-order staggered-grid FD operators. We verify the accuracy of the MOSN method using two test problems: a fixed-speed rupture case (Kostrov’s problem), and the case of a spontaneous rupture with slip-dependent friction. In the former test, comparisons of slip and shear stress waveforms confirm the convergence of numerical results to the analytical solution as the grid is refined. In the latter test, assessment is based on both qualitative waveform characteristics and a quantitative convergence analysis that uses error metrics in rupture time, final slip and peak slip rate. Compared with a previously verified second-order scheme, the MOSN method shows significant reduction in artefacts attributable to numerical dispersion and also yields a considerable improvement in rupture arrival times. While from a practical standpoint the performance gains of MOSN over the second-order scheme are considerable, MOSN has comparatively poorer convergence rates, and loses its advantage in accuracy in the limit of very small spatial steps. We attribute the relatively low convergence rates to our use of a friction law that induces gradient discontinuities in the fault traction, violating the smoothness assumptions underlying the accuracy estimates, a limitation which may not apply to more complex, experimentally derived friction laws with smooth dependence on slip-rate and state variables. Our current implementation is 2-D and assumes a straight fault, but the method is extensible to more general 3-D geometries.

In this work we perform an experimental study of iterative methods for solving large sparse linear systems arising from a second-order 2D mimetic discretization. The model problem is the 2D Poisson equation with different boundary conditions. We use GMRES with the restarted parameter and BiCGstab as iterative methods. We also use various preconditioning techniques including the robust preconditioner ILUt. The numerical experiments consist of large sparse linear systems with up to 643200 degrees of freedom.

We present an algorithm that reformulates existing methods to construct higher-order mimetic differential operators. Constrained linear optimization is the key idea of this resulting algorithm. The authors exemplified this algorithm by constructing an eight-order-accurate one-dimensional mimetic divergence operator. The algorithm computes the weights that impose the mimetic condition on the constructed operator. However, for higher orders, the computation of valid weights can only be achieved through this new algorithm. Specifically, we provide insights on the computational implementation of the proposed algorithm, and some results of its application in different test cases. Results show that for all of the proposed test cases, the proposed algorithm effectively solves the problem of computing valid weights, thus constructing higher-order mimetic operators.

Recent investigations have led to a method based on matrix analysis for the construction of high order discrete, fourth and higher mimetic operators, divergence and gradient. These operators have the same order of accuracy in the interior as well as at the boundary on uniform grids. This paper extends the method to non-uniform staggered grids. Here we present this extension on second-order operators although the method works for any order of accuracy. In addition, the boundary operator that allows the operators to satisfy a discrete analog of the divergence theorem is constructed.

Recent investigations by Castillo and Grone have led to a new method for constructing mimetic discretizations of divergence and gradient operators. Their technique, which employs a matrix formulation to incorporate mimetic constraints, is capable of producing approximations whose order at a grid boundary is equal to that in the grid’s interior. In this paper, we construct a second-order mimetic discretization using Castillo and Grone’s approach and compare it to other second-order discretizations by applying them to an elliptic boundary value problem in one dimension. A detailed perturbation analysis is provided to offer some insight into the two discretizations yielding the best numerical results in the study.

In situ hydrographic measurements at the entrance to the Gulf of California (GC) over the Alarcon Seamount (ASM) show that isopycnals and isotherms undulate, being displaced in an upward direction and generating a dome-like structure in the density, temperature and salinity fields, while only small perturbations in both fields are observed longitudinally. On the other hand, some along-gulf observations show an 80-m amplitude internal lee wave. We hypothesize that these perturbations result from the interaction of the mean flow with local topography, which generates internal lee waves as the flow passes over the ASM. Geostrophic velocities over the ASM (at 750 db level) show a meandering jet flow and two associated eddies. A fully three-dimensional boundary-fitted curvilinear ocean model in primitive variables is used to simulate and accurately represent the 3D circulation with high resolution in the neighborhood of the ASM. The simulated flow pattern shows a meandering jet over the topography with increased velocities on the flanks of the ASM. Interaction of current and topography generates a pair of baroclinic eddies similar to those observed during winter over the ASM. The associated density field consists of an upwelling on the upstream side of the seamount, leading to a patch of relatively dense water over the seamount and a downwelling on the lee side leading to a patch of relatively less dense water downstream.

We present a full 3-dimensional time dependent curvilinear ocean model in boundary-fitted grids. The model solves the primitive non-linear Navier-Stokes equations under the Boussinesq approximation in a rotating system. Three-dimensional simulations are made for a stratified flow over two different topographic configurations. The results for the velocity and density fields are presented and discussed.

An efficient and robust algorithm for numerical simulation of 2D stratified flow past an object is presented. The algorithm uses an accurate and efficient GMRES to solve the pressure. The algorithm is tested by simulating the flow past an infinite flat plate in uniform flow for various flow conditions. Results are presented for the density and velocity along with comparisons to the exact solution for this problem.

One-dimensional, second-order finite-difference approximations of the derivative are constructed which satisfy a global conservation law. Creating a second-order approximation away from the boundary is simple, but obtaining appropriate behavior near the boundary is difficult, even in one dimension on a uniform grid. In this article we exhibit techniques that allow the construction of discrete versions of the divergence and gradient operator that have high-order approximations at the boundary. We construct such discretizations in the one-dimensional situation which have fourth-order approximation both on the boundary and in the interior. The precision of the high-order mimetic schemes in this article is as high as possible at the boundary points (with respect to the bandwidth parameter). This guarantees an overall high order of accuracy. Furthermore, the method described for the calculation of the approximations uses matrix analysis to streamline the various mimetic conditions. This contributes to a marked clarity with respect to earlier approaches. This is a crucial preliminary step in creating higher-order approximations of the divergence and gradient for nonuniform grids in higher dimensions.

Numerical simulations of a linearly stratified diffusive flow past an impulsively started sphere are described. The impulsive start is perpendicular to isopycnal surfaces. The equations are transformed to generalized coordinates and then approximated by a finite difference scheme over a suitable grid which resolves both the diffusive and viscous boundary-layers. The results are compared with experimental evidence.

This paper incorporates variational, geometry and solution adaptive, grid-generation technology, with a new class of finite-volume methods to solve modeling problems with non-homogeneous and non-isotropic material properties. Current grid generation technology provides a powerful tool for adapting grids to complex geometries, discontinuities in the material properties, and discontinuities in the problem solution. The support-operators finite-volume methods have been proven to be effective in solving problems with discontinuous coefficients on general grids. The main goal of this research is to combine these two technologies to produce algorithms for fast and accurate solutions of difficult modeling problems.

When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations.

We introduce the divergence and the gradient for functions defined on a measure chain, and this includes as special cases both continuous derivatives and discrete forward differences. It is shown that in one dimension, subject to Dirichlet boundary conditions, the divergence and the gradient are negative adjoints of each other and that the divergence of the gradient is negative semidefinite. These are well-known results in tile continuous theory, and hence, mimic those properties also for the case of a general measure chain. @ 2001 Elsevier Science Ltd. All rights reserved.

We derive conservative fourth- and sixth-order finite difference approximations for the divergence and gradient operators and a compatible inner product on staggered 1D uniform grids in a bounded domain. The methods combine standard centered difference formulas in the interior with new one-sided finite difference approximations near the boundaries. We derive compatible inner products for these difference methods that are high-order approximations of the continuum inner product. We also investigate defining compatible high-order divergence and gradient finite difference operators that satisfy a discrete integration by parts identity.