Differences In Space Research Articles

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Tensor networks for solving the time-independent Boltzmann neutron transport equation

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing in space, (ii) multigroup-in-energy, and (iii) discrete ordinates collocation in angle leads to large generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores. We then solve the tensorized generalized eigenvalue problem using a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying the method to two examples of 3D neutron transport problems, currently solved by the Los Alamos National Laboratory PARallel TIme-dependent SN (PARTISN) solver.1 We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to large compression. This allows for the storage of terrabyte-sized neutron angular flux eigenvectors in megabytes. Additionally, we create megabyte-sized full access TT representations of yottabyte-sized transport matrix operators. By leveraging the TT operators and solution methods, we obtain a 7500 times speedup when compared to the PARTISN solution time with an error of less than 10−5.

Influence of Heat Accumulation on Morphology Debris Deposition and Wetting of LIPSS on Steel upon High Repetition Rate Femtosecond Pulses Irradiation

The fabrication of laser-induced periodic surface structures (LIPSS) over extended areas at high processing speeds requires the use of high repetition rate femtosecond lasers. It is known that industrially relevant materials such as steel experience heat accumulation when irradiated at repetition rates above some hundreds of kHz, and significant debris redeposition can take place. However, there are few studies on how the laser repetition rate influences both the debris deposition and the final LIPSS morphology. In this work, we present a study of fs laser-induced fabrication of low spatial frequency LIPSS (LSFL), with pulse repetition rates ranging from 10 kHz to 2 MHz on commercially available steel. The morphology of the laser-structured areas as well as the redeposited debris was characterized by scanning electron microscopy (SEM) and µ-Raman spectroscopy. To identify repetition rate ranges where heat accumulation is present during the irradiations, we developed a simple heat accumulation model that solves the heat equation in 1 dimension implementing a Forward differencing in Time and Central differencing in Space (FTCS) scheme. Contact angle measurements with water demonstrated the influence of heat accumulation and debris on the functional wetting behavior. The findings are directly relevant for the processing of metals using high repetition rate femtosecond lasers, enabling the identification of optimum conditions in terms of desired morphology, functionality, and throughput.

The feasibility of a 3D time-dependent model for predicting the area of possible sprite inception in the mesosphere based on an analytical solution to Poisson's equation

We present a formulation for calculating the electric field from the ground to the lower boundary of the ionosphere that is based on an analytical solution to Poisson's equation in three dimensions, combined with a numerical solution to the equation of charge continuity/conservation. This formulation allows one to follow the spatial and temporal evolution of the distribution of free charges and the electric field in the atmosphere and thus predict areas of possible sprite inception in the mesosphere in a quasi-electrostatic fashion for any thundercloud charge distribution. The main advantages of this formulation are: (1) for a given spatial resolution, the analytical solution to Poisson's equation is more numerically stable and accurate than a numerical solution to Poisson's equation that includes finite differencing in space; (2) unlike a numerical solution to Poisson's equation that includes finite differencing in space, the numerical stability is insensitive to the choice of spatial resolution; (3) no artificial side boundary condition need be applied; (4) no symmetry, cylindrical or otherwise, of the charge distributions nor the electric field distributions need be prescribed; and (5) the computation is readily parallelizable on multiple processors. The main limitation of the present formulation is that the electric field based on the electrostatic potential with upper and lower boundary conditions needs to be calculated for each charge in the domain, such that the larger the number of charges, the slower the computation. We explore the sensitivity of the formulation to its parameters, in order to elucidate its feasibility for performing simulations for a variety of thunderstorm charge distributions in 3D, and we demonstrate its utility in investigating the evolution of the area of possible sprite inception in cases of consecutive lightning discharges that are separated from one another in space as well as in time. In all of the simulations in the present study, the ambient electrical conductivity varies in space but is independent of time. However, as we discuss, our formulation can also accommodate a time-dependent conductivity profile.

SHOCK WAVES IN BIOLOGICAL TISSUES UNDER TELEGRAPH EQUATION HEAT CONDUCTION

Electrosurgery aimed at the removal of tumors results in highly transient heat conduction. Simulations of temperature fields, under the assumption that heat in soft tissue organs is governed by a telegraph (damped hyperbolic) equation, are conducted in a 2d setting with finite differencing in space and time. Six trajectories of a heat source motion are simulated: along a two-phase interface, along a two-phase interface with an offset, normal to a two-phase interface, on a curved (sine function) path in a single phase, along a circular path in a single phase, and along a circular path around inclusion of one type in a matrix of another type. With the surgeon’s hand motion velocity typically greater than the velocity of the telegraph equation’s second sound velocity, the heat source motion is supersonic, giving rise to evolving shock waves, Mach wedges, and high-temperature concentrations.

Simulating Electrochemical Behaviour of Lithium-Ion Cylindrical Cells Using Two-Dimensional Physics-Based Model

High energy storage density and energy conversion efficiency, rechargeability and zero tailpipe emissions of lithium-ion batteries make them an attractive choice for powering automobiles. However, significant research and improvement are still required to make these batteries comparable with the convenience associated with the use of fossil fuels. One of the major concerns associated with batteries in electric vehicles (EVs) is of safety due to the high current requirements. In this regard, battery pack designs can benefit tremendously by simulating adverse thermo-electrochemical scenarios. Due to the coupled nature of the electrochemical behaviour of the cell on its temperature distribution, the knowledge of temperature distribution and heat generation in constituting cells is required to predict the cell charge-discharge behaviour, local hot-spots formed within the cell and their influence on thermal runaways during pack operation. Various attempts towards this goal make use of several assumptions to simplify the cylindrical cell modelling. Lumped thermal models with constant temperature were used to determine the cell temperature directly using experimental data [1] or using the pseudo-2D model [2,3] on unwounded cells [4]. Heat generation was homogenized in the energy balance [5] and used along with experimental data for computation [6]. The lack of spatial-dependence in such models is not a good representation of physics expected in large format cylindrical cells due to their low effective transport and thermal properties, and the highly non-homogeneous material used to fabricate these cells.Some models that consider spatial dependence of physics employ this either in the one-dimensional radial direction [7], or the two-dimensional transverse section [8], or in the three-dimensions using two-dimensional resistor circuits for current collectors with one-dimensional resistors between adjacent current collector nodes [9]. For a proper understanding of local heat generation, the local three-dimensional coupled nature of thermo-electrochemical behaviour needs to be taken into account in the model. In a step towards this goal, the present work aims at developing a two-dimensional physics-based thermo-electrochemical model for cylindrical cells. The differential-algebraic governing equations have been solved implicitly with the non-linear term in each linearized for one variable using the Taylor series expansion. This approximation method ensures second-order accuracy for the linearized variable being computed while the non-linearity is retained for the other unknown variables. A variable time step solver based on the stability of the output voltage profile enables improved computational speed. A second-order differencing in space and backwards differencing in time have been implemented for discretizing the partial differential equations into algebraic equations. The Gauss-Seidel method is employed to iterate for the variables at each time step. This discretization technique has been validated (Fig.1a) over a range of C-rates (up to 6C) against an available solver [10]. Subsequently, the method is adapted for polar coordinates to simulate the two-dimensional cross-section of cylindrical cells. The discretization of the double-sided cylindrical cell geometry, as shown in Fig.1b, has been done ensuring nodes on all the boundaries separating various domains along the windings. The different domains within the coil have been discretized using Archimedean spirals with nodes at uniformly distributed angular coordinates. The number of tabs and their corresponding locations in the cell can be specified. The simulation results will be validated using data obtained from cycling of Samsung INR18650-29E lithium-ion cells on a high-current cell cycler in controlled temperature environment. In the future, the coupling of the electrochemical and thermal models for the cylindrical cell geometry will also be attempted.

Simulating Electrochemical Behaviour of Lithium-Ion Cylindrical Cells Using Two-Dimensional Physics-Based Model

High energy storage density and energy conversion efficiency, rechargeability and zero tailpipe emissions of lithium-ion batteries make them an attractive choice for powering automobiles. However, significant research and improvement are still required to make these batteries comparable with the convenience associated with the use of fossil fuels. One of the major concerns associated with batteries in electric vehicles (EVs) is of safety due to the high current requirements. In this regard, battery pack designs can benefit tremendously by simulating adverse thermo-electrochemical scenarios. Due to the coupled nature of the electrochemical behaviour of the cell on its temperature distribution, the knowledge of temperature distribution and heat generation in constituting cells is required to predict the cell charge-discharge behaviour, local hot-spots formed within the cell and their influence on thermal runaways during pack operation. Though many attempts were made in the past towards this goal, however, several assumptions were used to simplify the model for cylindrical cells. Lumped thermal models with constant temperature were used to determine the cell temperature directly using experimental data [1] or using pseudo-2D model [2,3] on unwounded cells [4]. Heat generation was homogenized in the energy balance [5] and used along with experimental data for computation [6]. The lack of spatial-dependence in such models is not a good representation of physics expected in large format cylindrical cells due to their low effective transport and thermal properties, and the highly non-homogeneous material used to fabricate these cells.Some models that consider spatial dependence of physics employ this either in one-dimensional radial direction [7], two-dimensional in transverse section [8], or three-dimensional approximation using two-dimensional resistor circuits for current collectors with non-linear one-dimensional resistors between adjacent current collector nodes [9]. For a proper understanding of local heat generation, the three-dimensional, highly local, and coupled nature of thermo-electrochemical behaviour needs to be taken into account in the model. In a step towards this goal, the present work aims at developing a two-dimensional physics-based thermo-electrochemical model for cylindrical cells. The differential-algebraic governing equations have been solved implicitly with the non-linear term in each linearized for one variable at a time using the Taylor series expansion. This approximation method ensures second-order accuracy for the linearized variable being computed while the non-linearity is retained for the other unknown variables. A variable time step solver based on the stability of the output voltage profile enables improved computational speed. A second-order differencing in space and backward differencing in time have been implemented for discretizing the partial differential equations into algebraic equations. The Gauss-Seidel method is employed to iterate for the variables at each time step. This discretization technique has been validated (Fig.1a) over a range of C-rates (up to 6C) against an available solver [10]. Subsequently, the method is adapted for polar coordinates to simulate the two-dimensional cross-section of cylindrical cells. The discretization of the geometry, as shown in Fig.1b, has been done ensuring nodes on all the boundaries separating various domains along the windings. The different domains within the coil have been discretized using Archimedean spirals with nodes at uniformly distributed angular coordinate. The simulation results will be validated using data obtained from cycling of Samsung INR18650-29E lithium-ion cells on a high-current cell cycler in controlled temperature environment. In the future, the coupling of the electrochemical and thermal models for the cylindrical cell geometry will also be attempted.

On the convergence of finite difference methods for PDE under temporal refinement

In this note we present an issue in the verification of order of convergence for finite difference approximations to partial differential equations discretized in time and space. When one refines both space and time together, the convergence rate for a correct implementation will match that expected from an analysis of the asymptotic convergence rate (or local truncation error) of the method. Sometimes, however, only the time-step size is reduced, while the spatial mesh is held fixed. The observed rate of convergence in this case may then differ from the formal order of accuracy of the method in space–time. In particular, one class of methods, second-order Lax–Wendroff time-differencing, produces only first-order when the time step is refined at fixed spatial mesh resolution because the time and space differencing are intrinsically linked. Our method to arrive at this result is through analyzing the error at a defined point in time associated with many time steps. We demonstrate our results computationally.

Numerical stability of explicit Runge–Kutta finite-difference schemes for the nonlinear Schrödinger equation

Linearized numerical stability bounds for solving the nonlinear time-dependent Schrödinger equation (NLSE) using explicit finite-differencing are shown. The bounds are computed for the fourth-order Runge–Kutta scheme in time and both second-order and fourth-order central differencing in space. Results are given for Dirichlet, modulus-squared Dirichlet, Laplacian-zero, and periodic boundary conditions for one, two, and three dimensions. Our approach is to use standard Runge–Kutta linear stability theory, treating the nonlinearity of the NLSE as a constant. The required bounds on the eigenvalues of the scheme matrices are found analytically when possible, and otherwise estimated using the Gershgorin circle theorem.

NLSEmagic: Nonlinear Schrödinger equation multi-dimensional Matlab-based GPU-accelerated integrators using compact high-order schemes

We present a simple to use, yet powerful code package called NLSEmagic to numerically integrate the nonlinear Schrödinger equation in one, two, and three dimensions. NLSEmagic is a high-order finite-difference code package which utilizes graphic processing unit (GPU) parallel architectures. The codes running on the GPU are many times faster than their serial counterparts, and are much cheaper to run than on standard parallel clusters. The codes are developed with usability and portability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with the MEX-compiler interface. The packages are freely distributed, including user manuals and set-up files. Program summaryProgram title: NLSEmagicCatalogue identifier: AEOJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEOJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 124453No. of bytes in distributed program, including test data, etc.: 4728604Distribution format: tar.gzProgramming language: C, CUDA, MATLAB.Computer: PC, MAC.Operating system: Windows, MacOS, Linux.Has the code been vectorized or parallelized?: Yes.Number of processors used: Single CPU, number of GPU processors dependent on chosen GPU card (max is currently 3072 cores on GeForce GTX 690).Supplementary material: Setup guide, Installation guide.RAM: Highly dependent on dimensionality and grid size. For typical medium–large problem size in three dimensions, 4GB is sufficient.Keywords: Nonlinear Schröodinger Equation, GPU, high-order finite difference, Bose-Einstien condensates.Classification: 4.3, 7.7.Nature of problem:Integrate solutions of the time-dependent one-, two-, and three-dimensional cubic nonlinear Schrödinger equation.Solution method:The integrators utilize a fully-explicit fourth-order Runge–Kutta scheme in time and both second- and fourth-order differencing in space. The integrators are written to run on NVIDIA GPUs and are interfaced with MATLAB including built-in visualization and analysis tools.Restrictions:The main restriction for the GPU integrators is the amount of RAM on the GPU as the code is currently only designed for running on a single GPU.Unusual features:Ability to visualize real-time simulations through the interaction of MATLAB and the compiled GPU integrators.Additional comments:Setup guide and Installation guide provided. Program has a dedicated web site at www.nlsemagic.com.Running time:A three-dimensional run with a grid dimension of 87×87×203 for 3360 time steps (100 non-dimensional time units) takes about one and a half minutes on a GeForce GTX 580 GPU card.

대도시 복합유역의 지표 및 지표하 유출해석기법 개발 (I)- 이론 및 모듈의 개발 -

Surface-subsurface interactions are an intrinsic component of the hydrologic response within a watershed. In general, these interactions are considered to be one of the most difficult areas of the discipline, particularly for the modeler who intends simulate the dynamic relations between these two major domains of the hydrological cycle. In essence, one major complexity is the spatial and temporal variations in the dynamically interacting system behavior. The proper simulation of these variations requires the need for providing an appropriate coupling mechanism between the surface and subsurface components of the system. In this study, an approach for modelling surface-subsurface flow and transport in a fully intergrated way is presented. The model uses the 2-dimensional diffusion wave equation for sheet surface water flow, and the Boussinesq equation with the Darcy`s law and Dupuit-Forchheimer`s assumption for variably saturated subsurface water flow. The coupled system of equations governing surface and subsurface flows is discretized using the finite volume method with central differencing in space and the Crank-Nicolson method in time. The interactions between surface and subsurface flows are considered mass balance based on the continuity conditions of pressure head and exchange flux. The major module consists of four sub-module (SUBFA, SFA, IA and NS module) is developed.

Maximum Norm a Posteriori Error Estimation For a Time-dependent Reaction-diffusion Problem

Abstract A semilinear second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for a difference scheme that uses Backward-Euler in time and central differencing in space. Sharp L¹-norm bounds for the Green's function of the parabolic operator and its derivatives are derived that form the basis of the a posteriori error analysis. Numerical results are presented.

Spatial Multigrid for Isotropic Neutron Transport

A spatial multigrid algorithm for isotropic neutron transport is presented in x-y geometry. The linear system is obtained using discrete ordinates in angle and corner balance finite differencing in space. Spatial smoothing is accomplished by a four-color block-Jacobi relaxation, where the diagonal blocks correspond to 4-cell blocks on the spatial grid. A bilinear interpolation operator and its transpose are used for the grid transfer operators. Good convergence factors were observed for homogeneous material properties. Heterogeneous material properties prove more difficult, especially the case of a vacuum region surrounded by a thick, diffusive region. In this case, a small amount of absorption, or “effective absorption” in a time-dependent problem, restores good convergence. Numerical results are presented.

An alternative discretization and solution procedure for the dual phase-lag equation

We describe an alternative numerical treatment of the dual phase-lag equation often used to account for microscale, short-time heat transport. The approach consists of an undecomposed formulation of the partial differential equation resulting from Taylor expansion with respect to lag times of the original delay partial differential equation. Trapezoidal integration in time and centered differencing in space provide an accurate discretization, as demonstrated by comparisons with analytical and experimental results in one dimension, and via grid-function convergence tests in three dimensions. For relatively fine 3-D grids the approach is approximately six times faster than a standard explicit scheme and nearly three times faster than an implicit method employing conjugate gradient iteration at each time step.

Black-hole excision with multiple grid patches

When using black-hole excision to numerically evolve a black-hole spacetime with no continuous symmetries, most 3 + 1 finite differencing codes use a Cartesian grid. It is difficult to do excision on such a grid because the natural r = constant excision surface must be approximated either by a very different shape such as a contained cube, or by an irregular and non-smooth ‘LEGO1 sphere’ which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme which uses multiple {r × (angular coordinates)} grid patches, each patch using a different (nonsingular) choice of angular coordinates. This allows excision on a smooth r = constant 2-sphere. I discuss the key design choices in such a multiple-patch scheme, including the choice of ghost-zone versus internal-boundary treatment of the interpatch boundaries (I use a ghost-zone scheme), the number and shape of the patches (I use a 6-patch ‘inflated-cube’ scheme), the details of how the ghost zones are ‘synchronized’ by interpolation from neighbouring patches, the tensor basis for the Einstein equations in each patch, and the handling of non-tensor field variables such as the BSSN (I use a scheme which requires ghost zones which are twice as wide for the BSSN conformal factor ϕ as for and the other BSSN field variables). I present sample numerical results from a prototype implementation of this scheme. This code simulates the time evolution of the (asymptotically flat) spacetime around a single (excised) black hole, using fourth-order finite differencing in space and time. Using Kerr initial data with J/m2 = 0.6, I present evolutions to t ⪆ 1500m. The lifetime of these evolutions appears to be limited only by outer boundary instabilities, not by any excision instabilities or by any problems inherent to the multiple-patch scheme.

A mathematical and numerical model for reactive fluid flow systems

We formulate mathematical and numerical models for multispecies, multi-phase and non-isothermal reactive fluid flow in porous media focusing on the chemical reactions and the transport of solutes. Mass conservation and stability in the time integration are emphasized. We use cell-centered finite volume differencing in space and an implicit Runge-Kutta method in time. Assuming two space dimensions, we introduce flux approximation for arbitrarily shaped convex quadrilaterals. On equidistant and variable sized rectangular grids we choose limited κ= $$\frac{1}{3}$$ related schemes to approximate the advective flux and the central difference scheme for the diffusive flux. On non-rectangular grids we recommend the VF9 scheme for the estimation of the diffusive flux. Our model exists as a code.

An economical second-order advection scheme for numerical weather prediction

A very simple second-order Eulerian scheme for the advection equation, based on a forward (backward) time integration on even (odd) grid points, is studied. the proposed scheme is similar, but not equivalent, to the so-called ‘hopscotch method’, developed in the 1960s and early 1970s for the advection-diffusion equation, and is stable up to Courant number 2.0. It is shown that, in the case of the advection equation, the proposed scheme has the same advantages yielded by the forward-backward scheme in the case of the shallow-water equations; in particular, it is equivalent to the application of centred time and space differencing on the Eliassen grid. the new scheme, unlike the classical leapfrog scheme, can be coupled to the forward-backward integration of the gravity-wave problem in primitive-equation models. With the aid of the proposed scheme, an explicit version of the atmospheric, Bologna Limited-Area Model (developed in recent years at the FISBAT Institute of the National Council of Research of Italy) is devised, and a comparison with the semi-implicit version of the same model is performed. the explicit version runs with a double time step, achieving the same accuracy with significantly less computer time and storage. It is suggested that the new time scheme is particularly suitable for numerical weather prediction on massively parallel computing machines based on SIMD (Single Instruction Multiple Data) and distributed memory architecture, which strongly penalize non-local algorithms.

An economical second‐order advection scheme for numerical weather prediction

AbstractA very simple second‐order Eulerian scheme for the advection equation, based on a forward (backward) time integration on even (odd) grid points, is studied. the proposed scheme is similar, but not equivalent, to the so‐called ‘hopscotch method’, developed in the 1960s and early 1970s for the advection‐diffusion equation, and is stable up to Courant number 2.0. It is shown that, in the case of the advection equation, the proposed scheme has the same advantages yielded by the forward‐backward scheme in the case of the shallow‐water equations; in particular, it is equivalent to the application of centred time and space differencing on the Eliassen grid. the new scheme, unlike the classical leapfrog scheme, can be coupled to the forward‐backward integration of the gravity‐wave problem in primitive‐equation models.With the aid of the proposed scheme, an explicit version of the atmospheric, Bologna Limited‐Area Model (developed in recent years at the FISBAT Institute of the National Council of Research of Italy) is devised, and a comparison with the semi‐implicit version of the same model is performed. the explicit version runs with a double time step, achieving the same accuracy with significantly less computer time and storage. It is suggested that the new time scheme is particularly suitable for numerical weather prediction on massively parallel computing machines based on SIMD (Single Instruction Multiple Data) and distributed memory architecture, which strongly penalize non‐local algorithms.

A two‐channel laterally averaged estuarine circulation model (LAECIM)

A two‐dimensional, two‐channel, laterally averaged estuarine circulation model (LAECIM) with a sigma coordinate in the vertical is developed for tides, tidal currents, and salinity in a two‐channel estuary with any arbitrary angle. The model is forced by oceanic tides composed of 16 major constituents and freshwater discharge. The two channels are governed by the same dynamic equations. A control volume at the junction between the two channels is designed to conserve mass and volume transport and to guarantee the continuity of momentum in the finite difference schemes, based on the physical principle in continuum medium of fluid. An explicit leapfrog scheme in time and centered differencing in space are employed, together with a time filter to remove the computational mode. An Euler backward scheme is periodically used to correct (retard) the phase frequency, which is overestimated due to the explicit leapfrog scheme used. The model is applied to the entire San Francisco Bay with its two channels: the main north‐south channel and the Golden Gate channel, which is perpendicular to the former (i.e., with a right angle to the former). The simulated time series of tides, tidal currents, and salinities compare favorably with the observed tidal gauge data and moored current and salinity time series during the summer 1980. The simulated along‐channel salinity section also compares well to the conductivity‐temperature‐depth sectional observations during the same period. The model simulated Eulerian residual current is consistent with two‐layer flow theory.

Spherical symmetric diffusion problem

We introduce a general and versatile MS Windows application for solving the spherically symmetric diffusion problem, involving up to two coupled spherically symmetric Smoluchowski equations. The application is based on a modular, configurable, user-friendly graphical interface, in which input parameters are introduced through a graphical representation of the system of partial differential equations and output attributes can be obtained graphically during propagation. The numerical algorithm consists of finite differencing in space and Chebyshev propagation in time; it includes an implementation of virtual gridding, which enhances the accuracy of calculating boundary conditions and steep potentials. The program has b een checked against a wide collection of analytical solutions and applied to an experimentally open problem in excited-state proton-transfer to solvent. © 1996 by John Wiley & Sons, Inc.