Explicit Numerical Scheme Research Articles

An accurate numerical solution algorithm for 3D velocity–vorticity Navier–Stokes equations by the DQ method

AbstractA numerical scheme based on a differential quadrature (DQ) method is discussed for the solution of three‐dimensional velocity–vorticity Navier–Stokes equations. Numerical solutions of the six field variables, three velocities and three vorticities are obtained by adopting a coupled numerical solution procedure. Hence, it is required to specify only the velocity boundary conditions, whereas the vorticity values at the boundary are computed implicitly, thus without seeking boundary vorticity values by an iterative procedure followed in other explicit numerical schemes. Since the DQ method approximates a given differential operator with higher‐order polynomials, the vorticity definition is accurately enforced at the boundary. This ensures a divergence‐free solution for the flow field. Flow simulations obtained for a lid‐driven cubic cavity for Stokes flow and Re=100, 400 and 1000 show excellent agreements with other numerical schemes. A grid independence study demonstrates that the present numerical scheme could predict benchmark solutions with coarser meshes than other numerical schemes. Copyright © 2005 John Wiley & Sons, Ltd.

On Discretization Schemes for Stochastic Evolution Equations

Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.

Finite Element Methods for Parabolic Stochastic PDE?s

We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include the forward and backward Euler and the Crank–Nicholson schemes. We use the finite element method. We find, as expected, that the rates of convergence are substantially similar to those found for finite difference schemes, at least when the size of the time step k is on the order of the square of the size of the space step h: all the schemes considered converge at a rate on the order of h1/2+k1/4, which is known to be optimal. We also consider cases where k is much greater than h2, and find that only the backward Euler method always attains the optimal rate; other schemes, even though they are stable, can fail to convergence to the true solution if the time step is too long relative to the space step. The Crank–Nicholson scheme behaves particularly badly in this case, even though it is a higher-order method.

Integrating legacy components into a software system for storm sewer simulation

This paper presents an approach that integrates a legacy component into a software system for storm sewer simulation. The legacy component employed here is the Storm Water Management Model (SWMM). The Extended Transport (EXTRAN) block of the SWMM that applies the finite difference method (FDM) with explicit numerical schemes, solving the de Saint-Venant equations, is used to route the storm sewer flow. A storm sewer simulation system, named S4, that integrates SWMM-EXTRAN and implements a visualization model, has been developed to demonstrate the proposed approach. The approach makes use of the multi-thread technology to alternate the execution between SWMM-EXTRAN for flow simulation on one thread and the program controller that updates simulation state variables and displays the computed temporal water-stages at the junctions on the other thread at every time step of the FDM process. Two test examples are used to verify and demonstrate the feasibility of the proposed approach. The results show that the multi-thread technology is applied successfully for integrating legacy components, such as SWMM-EXTRAN, into a software system (in this case, S4). In addition, the proposed approach is generally applicable for integrating legacy models or components developed using FDM with explicit numerical schemes.

Modifications of the Helbing-Molnár-Farkas-Vicsek Social Force Model for Pedestrian Evolution

A model of crowd motion that considers each pedestrian as a Newtonian particle subject to both physical and social forces was reported by Helbing, Farkas, and Vicsek in 2000. Subsequent numerical simulations of this model, performed by its authors, showed that it exhibits realistic crowd behavior. In this article, the authors point out that numerical values of certain parameters in that model may produce counterintuitive results when applied to the motion of an isolated pedestrian or a small number of pedestrians. They have considered modifications of the original model, which allow them to use parameter values that, in the aforementioned sense, are more realistic. However, this is achieved by introducing more features and parameters into the original model. These features are described, and some results of the numerical simulations of the modified model are presented. Two major results of their study need to be mentioned. First, they developed an algorithm, based on an explicit numerical integration scheme, which prevents simulated pedestrians from overlapping with one another in physical space. Second, they demonstrated how the form of the social repulsive force between two pedestrians may be deduced from certain measured characteristics of pedestrian flows.

Modelling estuarine and coastal flows using an unstructured triangular finite volume algorithm

Details are given of the development and application of a numerical model for predicting free-surface flows in estuarine and coastal basins using the finite volume method. Both second- and third-order accurate and oscillation free explicit numerical schemes have been used to solve the shallow water equations. The model deploys an unstructured triangular mesh and incorporates two types of mesh layouts, namely the ‘cell centred’ and ‘mesh vertex’ layouts, and provides a powerful mesh generator in which a user can adjust the mesh-size distribution interactively to create a desirable mesh. The quality of mesh has been shown to have a major impact on the overall performance of the numerical model. The model has been applied to simulate two-dimensional dam break flows for which transient water level distributions measured within a laboratory flume were available. In total 12 model runs were undertaken to test the model for various flow conditions. These conditions include: (1) different bed slopes (ranging from zero to 0.8%), (2) different upstream and downstream water level conditions, and (3) initially wet and dry bed conditions, downstream of the dam. Detailed comparisons have been made between model predicted and measured water levels and good agreement achieved between both sets of results. The model was then used to predict water level and velocity distributions in a real estuary, i.e. the Ribble Estuary, where the bed level varies rapidly at certain locations. In order to model the whole estuary, a 1-D numerical model has also been used to model the upper part of the estuary and this model was linked dynamically to the 2-D model. Findings from this application are given in detail.

Numerical modelling for earthquake engineering: the case of lightly RC structural walls

AbstractDifferent types of numerical models exist to describe the non‐linear behaviour of reinforced concrete structures. Based on the level of discretization they are often classified as refined or simplified ones. The efficiency of two simplified models using beam elements and damage mechanics in describing the global and local behaviour of lightly reinforced concrete structural walls subjected to seismic loadings is investigated in this paper. The first model uses an implicit and the second an explicit numerical scheme. For each case, the results of the CAMUS 2000 experimental programme are used to validate the approaches. Copyright © 2004 John Wiley & Sons, Ltd.

A fast multi-scale edge detection algorithm

In this paper we present a new explicit numerical scheme to approximate the solution of the linear diffusion filtering. This scheme is fast, stable, easy to program, applicable to arbitrary dimensions, and preserves the discontinuities of the objects. Experimental results support the efficiency of the proposed approach for the multi-scale detection of edges in greyscale, and color images.

Application of the Conservation Element and Solution Element Method in Numerical Modeling of Axisymmetric Heat Conduction with Melting and/or Freezing

The space-time conservation element and solution element (CE/SE) method, an accurate and efficient explicit numerical scheme for resolving moving discontinuities in fluid mechanics problems, is used to solve axisymmetric heat conduction problems with melting and/or freezing. The axisymmetric formulation is presented. Comparisons are made to existing analytical solutions. The CE/SE method is found to be accurate and robust for the numerical modeling of phase change problems.

A multiprocessor coupled ice‐ocean model for the Baltic Sea: Application to salt inflow

Within the Swedish Regional Climate Modeling Program, SWECLIM, a three‐dimensional (3‐D) coupled ice‐ocean model for the Baltic Sea has been developed to simulate physical processes on timescales of hours to decades. The code has been developed based on the massively parallel version of the Ocean Circulation Climate Advanced Modeling (OCCAM) project of the Bryan‐Cox‐Semtner model. An elastic‐viscous‐plastic ice rheology is employed, resulting in a fully explicit numerical scheme that improves computational efficiency. An improved two‐equation turbulence model has been embedded to simulate the seasonal cycle of surface mixed layer depths as well as deepwater mixing on decadal timescale. The model has open boundaries in the northern Kattegat and is forced with realistic atmospheric fields and river runoff. Optimized computational performance and advanced algorithms to calculate processor maps make the code fast and suitable for multi‐year, high‐resolution simulations. As test cases, the major salt water inflow event in January 1993 and the stagnation period 1980–1992, have been selected. The agreement between model results and observations is regarded as good. Especially, the time evolution of the halocline in the Baltic proper is realistically simulated also for the longer period without flux correction, data assimilation, or reinitialization. However, in particular, smaller salt water inflows into the Bornholm Basin are underestimated, independent of the horizontal model resolution used. It is suggested that the mixing parameterization still needs improvements. In addition, a series of process studies of the inflow period 1992/1993 have been performed to show the impact of river runoff, wind speed, and sea level in Kattegat. Natural interannual runoff variations control salt water inflows into the Bornholm Basin effectively. The effect of wind speed variation on the salt water flux from the Arkona Basin to the Bornholm Basin is minor.

Advection Tests of Optimal Compact Implicit Scheme

Gross, Koseff, and Monismith used two advection test cases to expose the poor performance of seven widely used explicit numerical schemes. Here, those test cases are used to reveal the weakness and strength of the optimal compact implicit scheme. In the diagonal advection of a sharp square block, there are sawtooth grid scale errors. Binomial smoothing, with standard deviation of one grid spacing of the initial square block is required before the optimal compact implicit scheme improves upon the best of the explicit schemes. For the circular advection of a smooth narrow Gaussian cone, the accuracy is 105 times better than the best of the explicit schemes.

Review Paper: Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations

In recent years, after a prolonged period of intense activity, the study of numerical methods for solving stiff initial-value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity. There now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems. In this paper, we sketch some of the main theory which underpins stiff integration methods and we use this to describe, and put into context, some of the best codes currently available. By referencing only codes which have been thoroughly tested and are widely available, our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type. An additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated.

The third‐order polynomial method for two‐dimensional convection and diffusion

AbstractUsing the upstream polynomial approximation a series of accurate two‐dimensional explicit numerical schemes is developed for the solution of the convection–diffusion equation. A third‐order polynomial approximation (TOP) of the convection term and a consistent second‐order approximation of the diffusion term are combined in a single‐step flux‐difference algorithm. Stability analysis confirms that the TOP‐12 scheme satisfies the CFL condition for two dimensions. Using smaller and narrower flux stencils compared to algorithms of similar accuracy, the TOP‐12 scheme is more efficient in terms of computations per single node. Numerical tests and comparison with other well‐known algorithms show a high performance of the developed schemes. Copyright © 2003 John Wiley & Sons, Ltd.

Enhanced and restored signals as a generalized solution for shock filter models. Part II—numerical study

In Part I of this paper, we proposed a well-posed generalized model for signal enhancement and restoration based on shock filters. A theoretical study of the Cauchy problem in the framework of generalized functions algebra was developed in detail. In Part II, we investigate the numerical aspects of the model. We derive an efficient, explicit numerical scheme in both one and two dimensions, and investigate the schemes' stability and convergence. Through experimental tests, we demonstrate the effectiveness of the numerical schemes when restoring and enhancing signals in various situations with a limited number of iterations. Moreover, we show the impact of the coefficients introduced in the model on the procedure's processing time.

Numerical Approximation of the Lifshitz--Slyozov--Wagner Equation

The Lifshitz--Slyozov--Wagner theory of coarsening (Ostwald ripening) describes the late stages of the growth by diffusional mass transfer of the grains of a new phase from a supersaturated solution. It results in a nonlinear transport equation with a nonlocal nonlinearity for the volume distribution function of the grains. A time explicit finite volume numerical scheme is proposed to solve this equation in self-similar variables and is shown to converge under a CFL condition. Numerical simulations are also presented.

Numerical simulation of morphological changes in rivers and reservoirs

For the computation of the morphological changes in river or reservoir beds, due to erosion or deposition, a hydrodynamic model is combined with the sediment continuity equation. The hydrodynamic model is based upon the equations of mass and momentum conservation. For the solution of the above system of three partial differential equations, the explicit numerical schemes of MacCormack and Lax-Wendroff are tested. The whole mathematical model is applied to two experimental cases and to a real case. The first experimental case concerns the aggradation in a laboratory channel due to sediment overloading, and the second one the sediment release from a reservoir after a dam break. The real case concerns the morphological changes in the torrent Mallero (Northern Italy) during a flood event.

Explicit Schemes for Dam-Break Simulations

Dam-break problems involve the formation of shocks and rarefaction fans. The performance of 20 explicit numerical schemes used to solve the shallow water wave equations for simulating the dam-break problem is examined. Results from these schemes have been compared with analytical solutions to the dam-break problem with finite water depth and dry bed downstream of the dam. Most of the numerical schemes produce reasonable results for subcritical flows. Their performance for problems where there is a transition between subcritical and supercritical flows is mixed. Although many numerical schemes satisfy the Rankine-Hugoniot condition, some produce solutions which do not satisfy the entropy condition, producing nonphysical solutions. This was the case for the majority of first-order schemes examined. Numerical schemes which consider critical flow in the solution are guaranteed to produce entropy satisfying solutions. Second-order schemes avoid the generation of expansive shocks; however, some form of flux or slope limiter must be used to eliminate oscillations that are associated with these schemes. These limiters increase the complexity and the computational effort required, but they are generally more accurate than their first-order counterparts. The limiters employed by these second-order schemes will produce monotone or total variation diminishing solutions for scalar equations. Some limiters do not exhibit these properties when they are applied to the nonlinear shallow water wave equations. This comparative study shows that there are a variety of shock-capturing numerical schemes that are efficient, accurate, robust, and are suitable for solving the shallow water wave equations when discontinuities are encountered in the problem.

Advanced Dynamics

7R7. Advanced Dynamics. - RB Bhat (Concordia Univ, Montreal, Canada) and RV Dukkipati (Sch of Eng, Fairfield Univ, Fairfield CT 06430-5195). Narosa Publ, New Delhi, India. Distributed in USA by CRC Press LLC, Boca Raton FL. 2001. 395 pp. ISBN 0-8493-1018-0. $99.95.Reviewed by M Pascal (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Paris, 75252 Cedex 05, France).The authors’ aim is to provide a textbook giving insight about the methods used in Analytical Mechanics and which are related to abstract concepts like generalized parameters, virtual displacements, and variational principles. In contrast with other textbooks, the authors did not assume that the basic knowledge of kinematics and dynamics is familiar to the reader; the transition to this abstract presentation of the fundamental laws of mechanics is performed very gradually. The result is a book involving altogether topics related to elementary backgrounds (suitable for first-year graduate students) about dynamics of particles and rigid bodies and advanced theoretical part (useful for undergraduate senior students). The book involves 12 chapters, each of them is followed by a list of references and a great amount of homework problems. The fundamental laws of Mechanics are presented in the first chapter. Only systems of particles or plane motions of rigid bodies are considered in this chapter. Several topics (kinematics of rigid bodies, radius of gyration, center of percussion…) are introduced, but some other basic concepts like Galileen frame, relative and absolute velocities, Coriolis acceleration…are missing and will be presented only in Chapter 4. This reviewer is not convinced that this first chapter is very suitable. For first level students, a more detailed presentation is needed, and for senior students, all these concepts are known. Chapter 2 deals with more interesting topics such as generalized coordinates, classification of constraints, virtual works principle, and generalized forces. However, in this part, only systems of particles are considered, and the fundamental distinction between external and internal forces is not underlined. Several elementary concepts are presented in Chapters 3 and 4, like potential and kinetics energies, linear and angular momentum vectors, relative motions, and transformation of coordinates systems. The Foucault’s pendulum gives an interesting example of these topics. In Chapter 5, an introduction to Kepler’s laws and to orbital motion is presented. Chapter 6 deals with Lagrange equations for holonomic and non-holonomic systems. First integrals are introduced for systems with cyclic coordinates. A great amount of examples are solved. The next two chapters are devoted to basic knowledges about inertia tensors, principal axes of inertia and dynamics of rigid bodies. As an example, the motion of the spinning top is solved. Chapter 9 presents variational methods. Hamilton’s and the least action principle are derived, together with Hamilton’s equations. Canonical transformations and Hamilton-Jacobi equations are studied in Chapter 10; Lagrange and Poisson brackets are also introduced.An interesting classification of dynamics systems is given at the beginning of Chapter 11, devoted to vibrations. Then the linear vibrations of single-degree-of-freedom systems are investigated. The forced vibrations are also considered for harmonic and arbitrary excitation forces. For multi-degree-of-freedom systems, usual modal analysis is presented for undamped and damped cases. The last chapter provides an introduction to numerical methods. Several explicit and implicit numerical schemes are presented for the integration of linear dynamical systems. An example of a non-linear system is then solved by means of these numerical methods in order to underline the problems encountered in this case. In conclusion, this reviewer thinks that this book is without doubt very useful at a time when many students, and even many scientists, think that Analytical Mechanics belongs to the past. Several interesting examples of the use of these methods may be found in this book. In spite of rather numerous printing mistakes in the text, the book is clearly written with a lot of good quality figures. This reviewer thinks that Advanced Dynamics can be used by senior students and also scientists involved in systems dynamics and interested in analytical methods rather than numerical methods.

Performance Analysis of a Multiprocessor Coupled Ice–Ocean Model for the Baltic Sea

Within the Swedish Regional Climate Modelling Programme (SWECLIM) a 3D coupled ice-ocean model for the Baltic Sea has been developed to simulate physical processes on timescales of hours to decades. The model code is based on the global ocean GCM of the Ocean Circulation Climate Advanced Modelling (OCCAM) project and has been optimized for massively parallel computer architectures. The Hibler-type dynamic-thermodynamic sea ice model utilizes elastic-viscous-plastic rheology resulting in a fully explicit numerical scheme that improves computational efficiency. A detailed performance analysis shows that the ice model causes generic workload imbalance between involved processors. An improved domain partitioning technique minimizes load imbalance, but cannot solve the problem completely. However, it is shown that the total load imbalance is not more than 13% for a mild winter and about 8% for a severe winter. With respect to parallel processor performance, the code makes the best use of available computer resources.

3D Micromechanical modeling of packed beds

A new 3D numerical model has been developed to simulate the thermal and mechanical characteristics of packed beds used in fusion reactor blankets and other applications. This method is based on an explicit numerical scheme which monitors the interaction of the particles contact by contact and their motion particle by particle. In this paper, a mathematical formulation as well as a model which predicts the packed bed thermomechanical states under imposed and induced loads are presented. The model is validated by comparing the numerical simulations with thermal expansion and uniaxial compression experiments of packed beds. The results of the calculated bed effective modulus compares reasonably well with the experimental data.