Practical Aspects in Comparing Shock-Capturing Schemes for Dam Break Problems

The results of a survey aimed at comparing the performances of first-order and total variation diminishing (TVD) second-order upwind flux difference splitting schemes, first-order space-centered schemes, and second-order space-centered schemes with the TVD artificial viscosity term are reported here. The schemes were applied to the following dam-break wave cases: in a dry frictionless horizontal channel; in a dry, rough and sloping channel; and in a nonprismatic channel. Among first-order schemes, the diffusive scheme provides only slightly less accurate results than those obtained by the Roe scheme. For TVD second-order schemes, no significant difference between the upwind scheme and central schemes are reported. In the case of a dam break in a dry frictionless horizontal channel, the second-order schemes were two- to five-fold more accurate than the diffusive scheme and Roe’s scheme. These differences in scheme performances drastically reduce when the results obtained for the rough sloping channel test and for the nonprismatic channel test are analyzed. In particular, the accuracy of the diffusive and Roe’s schemes is similar to second-order schemes when such features of dam break wave, relevant from an engineering viewpoint, like wave peak arrival time and maximum water depths, are considered.     

Godunov-Type Solutions for Water Hammer Flows

First- and second-order explicit finite volume (FV) Godunov-type schemes for water hammer problems are formulated, applied, and analyzed. The FV formulation ensures that both schemes conserve mass and momentum and produce physically realizable shock fronts. The exact solution of the Riemann problem provides the fluxes at the cell interfaces. It is through the exact Riemann solution that the physics of water hammer waves is incorporated into the proposed schemes. The implementation of boundary conditions, such as valves, pipe junctions, and reservoirs, within the Godunov approach is similar to that of the method of characteristics (MOC) approach. The schemes are applied to a system consisting of a reservoir, a pipe, and a valve and to a system consisting of a reservoir, two pipes in series, and a valve. The computations are carried out for various Courant numbers and the energy norm is used to evaluate the numerical accuracy of the schemes. Numerical tests and theoretical analysis show that the first-order Godunov scheme is identical to the MOC scheme with space-line interpolation. It is also found that, for a given level of accuracy and using the same computer, the second-order scheme requires much less memory storage and execution time than either the first-order scheme or the MOC scheme with space-line interpolation. Overall, the second-order Godunov scheme is simple to implement, accurate, efficient, conservative, and stable for Courant number less than or equal to one.     

Impact of Limiters on Accuracy of High-Resolution Flow and Transport Models

High-resolution finite volume schemes used to predict mass transport and free surface flows utilize limiters such as Minmod, Double Minmod, and Superbee to prevent spurious oscillations commonly associated with second-order accurate schemes. These limiters effectively switch between the classical Lax-Wendroff, Warming-Beam, and Fromm schemes, or amplified versions of these schemes that artificially increase gradient magnitudes to minimize damping of high frequency solution components. A Von Neumann analysis illustrates that gradient or slope amplification reduces numerical dissipation, but also increases the phase error and should therefore be cautiously used. The Double Minmod limiter closely mimics the Fromm scheme and possesses better phase accuracy than the Minmod and Superbee limiters. Near sharp solution gradients, slope amplification used by the Double Minmod and Superbee limiters reduces artificial smearing. The Minmod limiter does not use slope amplification and therefore yields the most solution smearing. Results of model tests show that the combined attributes of the Double Minmod limiter yield more accurate predictions of mass transport and circulation zones in shallow water than those of other limiters such as Minmod and Superbee.     

Flux-Difference Splitting Schemes for 2D Flood Flows

Two numerical models for 2D flood flows are presented. One model is first-order accurate and another is second-order accurate. Roe's numerical flux is used to develop the first-order accurate model, while second-order accuracy, in space and time, is obtained by using the Lax-Wendroff numerical flux. A simple operator splitting is found to yield the same results as that obtained by using more complicated, and thus, time consuming, operator splitting. Roe's approximate Jacobian is used for conservative properties and Harten and Hyman's procedure is followed for the entropy inequality condition. Flux limiter is used in the second-order accurate model that removes oscillations while maintaining the order of accuracy. The models are verified against available experimental data of a 2D flood wave due to partial dam-break. Numerical experiments are conducted to verify the models' ability to correctly predict behavior of the free surface, in addition to prediction of depth and velocity.     

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.     

Finite Volume Model for Shallow Water Equations with Improved Treatment of Source Terms

A simple yet precise relation between the flux gradient and the bed slope source term is presented, which produces a net force within the cell with an inclined water surface, but ensures still water condition when there is no flow across the boundaries. The proposed method consists of calculating the pressure term based on the water depths at the cell vertices, which may be computed by a higher order scheme and the bed slope source term by a centered discretization technique. The methodology is demonstrated with a Godunov-type upwind finite volume formulation. The inviscid fluxes are calculated using Roe’s approximate Riemann solver and a second-order spatial accuracy is obtained by implementing multidimensional gradient reconstruction and slope limiting techniques. The accuracy and applicability of the numerical model is verified with a couple of test problems and a real flow example of tidal water movement in a stretch of River Hooghly in India.     

Discretization of Integral Equations Describing Flow in Nonprismatic Channels with Uneven Beds

Application of the finite-volume method in one dimension for open channel flow predictions mandates the direct discretization of integral equations for mass conservation and momentum balance. The integral equations include source terms that account for the forces due to changes in bed elevation and channel width, and an exact expression for these source term integrals is presented for the case of a trapezoidal channel cross section whereby the bed elevation, bottom width, and inverse side slope are defined at cell faces and assumed to vary linearly and uniformly within each cell, consistent with a second-order accurate solution. The expressions may be used in the context of any second-order accurate finite-volume scheme with channel properties defined at cell faces, and it is used here in the context of the Monotone Upwind Scheme for Conservation Laws (MUSCL)-Hancock scheme which has been adopted by many researchers. Using these source term expressions, the MUSCL-Hancock scheme is shown to preserve stationarity, accurately converge to the steady state in a frictionless flow test problem, and perform well in field applications without the need for upwinding procedures previously reported in the literature. For most applications, an approximate, point-wise treatment of the bed slope and nonprismatic source terms can be used instead of the exact expression and, in contrast to reports on other finite-volume-based schemes, will not cause unphysical oscillations in the solution.     

Finite-Difference TVD Scheme for Computation of Dam-Break Problems

A second-order hybrid type of total variation diminishing (TVD) finite-difference scheme is investigated for solving dam-break problems. The scheme is based upon the first-order upwind scheme and the second-order Lax-Wendroff scheme, together with the one-parameter limiter or two-parameter limiter. A comparative study of the scheme with different limiters applied to the Saint Venant equations for 1D dam-break waves in wet bed and dry bed cases shows some differences in numerical performance. An optimum-selected limiter is obtained. The present scheme is extended to the 2D shallow water equations by using an operator-splitting technique, which is validated by comparing the present results with the published results, and good agreement is achieved in the case of a partial dam-break simulation. Predictions of complex dam-break bores, including the reflection and interactions for 1D problems and the diffraction with a rectangular cylinder barrier for a 2D problem, are further implemented. The effects of bed slope, bottom friction, and depth ratio of tailwater∕reservoir are discussed simultaneously.     

Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme

A finite volume well-balanced weighted essentially nonoscillatory (WENO) scheme, fourth-order accurate in space and time, for the numerical integration of shallow water equations with the bottom slope source term, is presented. The main novelty introduced in this work is a new method for managing bed discontinuities. This method is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Further changes regard the treatment of the source term, based on a high-order extension of the divergence form for bed slope source term method, and the application of an analytical inversion of the specific energy-depth relationship. Two ad hoc test cases, consisting of a steady flow over a step and a surge crossing a step, show the effectiveness of the method of treating bottom discontinuities. Several standard one-dimensional test cases are also used to verify the high-order accuracy, the C-property, and the good resolution properties of the resulting scheme, in the cases of both continuous and discontinuous bottoms. Finally, a comparison between the fourth-order scheme proposed here and a well-established second-order scheme emphasizes the improvement achieved using the higher-order approach.     

Investigation of the Importance of Spatial Resolution for Two-Dimensional Shallow-Water Model Accuracy

The importance of spatial resolution for two-dimensional shallow-water model accuracy has been investigated by testing the effect of mesh refinement on two test cases based on laboratory dam-break experiments. A balanced first-order accurate upwind Q-Scheme and a second-order accurate upwind Hancock Monotone Upstream-centered Scheme for Conservation Laws scheme were both first validated on an analytical test, and then applied to the experimental dam-break test cases on four meshes of different density. Simulation results were evaluated through comparison of experimental and computed water level values at several available gauge points. Model sensitivity analysis showed that (1) mesh density was not critical for results accuracy; (2) excessive mesh refinement somewhat deteriorated the results; and (3) optimal spatial resolution was relatively low. Response is shown to be highly complex and no simple relation between spatial resolution and model accuracy has been found.     

Unstructured Grid Finite-Volume Algorithm for Shallow-Water Flow and Scalar Transport with Wetting and Drying

A high-resolution, unstructured grid, finite-volume algorithm is developed for unsteady, two-dimensional, shallow-water flow and scalar transport over arbitrary topography with wetting and drying. The algorithm uses a grid of triangular cells to facilitate grid generation and localized refinement when modeling natural waterways. The algorithm uses Roe’s approximate Riemann solver to compute fluxes, a multidimensional limiter for second-order spatial accuracy, and predictor–corrector time stepping for second-order temporal accuracy. The novel aspect of the algorithm is a robust and efficient procedure to consistently track fluid volume and the free surface elevation in partially submerged cells. This leads to perfect conservation of both fluid and dissolved mass, preservation of stationarity, and near elimination of artificial concentration and dilution of scalars at stationary or moving wet/dry interfaces. Multi-dimensional slope limiters, variable reconstruction, and flux evaluation schemes are optimized in the algorithm on the basis of accuracy per computational effort.     

Godunov-Type Solutions for Transient Flows in Sewers

This work is part of a long term project which aims at developing a hydraulic model for real-time simulation of unsteady flows in sewers ranging from gravity flows, to partly gravity–partly surcharged flows to fully surcharged flows. The success of this project hinges on the ability of the hydraulic model to handle a wide range of complex boundaries and to provide accurate solutions with the least central processing unit time. This first paper focuses on the development and assessment of two second-order explicit finite-volume Godunov-type schemes (GTS) for unsteady gravity flows in sewers, but with no surcharging. Traditionally, hydraulic transients have been modeled using the method of characteristics (MOC), which is noted for its ability to handle complex boundary conditions (BCs). The two GTS described herein incorporate BCs in a similar manner to the MOC. The accuracy and efficiency of these GTS schemes are investigated using problems whose solution contains features that are relevant to transient flows in sewers such as shock, expansion, and roll waves. The results show that these GTS schemes are significantly faster to execute than the fixed-grid MOC scheme with space-line interpolation, and in some cases, the accuracy produced by the two GTS schemes cannot be matched by the accuracy of the MOC scheme, even when a Courant number close to one and a large number of grids is used. Furthermore, unlike the MOC solutions, which exhibit increasing numerical dissipation with decreasing Courant numbers, the resolution of the shock fronts was maintained by the GTS schemes even for very low Courant numbers (0.001).     

基于逆控制的永磁同步电机速度伺服系统

针对永磁同步电机系统解耦性能,提出一种逆控制的新型控制策略,应用逆系统方法对永磁同步电机进行解耦控制研究.通过对永磁同步电机的数学模型可逆性分析,将永磁同步电机系统解耦成二阶线性转速与一阶线性定子电流两个低阶的线性子系统.仿真结果表明,控制方案具有优良的动态和静态性能,且对负载变化具有较强鲁棒性.     

Adaptive Parametric Identification Scheme for a Class of Nondeteriorating and Deteriorating Nonlinear Hysteretic Behavior

The adaptive parametric identification of deteriorating and nondeteriorating nonlinear hysteretic phenomena is considered using a generalization of Masing model based on the observed memory behavior of distributed element models. The model permits a parametric identification to be performed using nonlinear optimization techniques for arbitrary response time histories. A changing objective function, defined as the normalized force estimation error over a shifting window of recent data, is employed so that classic nonlinear optimization techniques can be used for the adaptive identification problem. A variation of the steepest descent method is used with significant modifications. To achieve the best performance for any given problem, a set of a priori numeric tests are suggested to design the identification scheme. The design identification scheme exhibits a very good performance in identifying the correct values of the parameters and is rather robust in dealing with noise. The proposed approach has applications to adaptive identification of much wider types of nonlinear rate-dependent hysteretic behavior. Also, the set of guidelines proposed by the authors is a contribution toward having more effective autonomous identification schemes, using minimal information about the model and input.     

Accuracy Order of Crank–Nicolson Discretization for Hydrostatic Free-Surface Flow

Application of Crank–Nicolson (CN) discretization to the hydrostatic (or shallow-water) free-surface equation in two-dimensional or three-dimensional Reynolds-averaged Navier–Stokes models neglects a second order term. The neglected term is zero at steady state, so it does not appear in steady-state accuracy analyses. A new correction term is derived that restores second-order accuracy. The correction is significant when the amplitude of the surface oscillation is within two orders of magnitude of the water depth and the barotropic Courant–Friedrichs–Lewy (CFL) stability condition is less than unity. Analysis shows that the CN accuracy for an unforced free-surface oscillation is degraded to first order when the barotropic CFL stability condition is greater than unity, independent of whether or not the new correction term is applied. The results indicate that the semi-implicit Crank–Nicolson method, applied to the hydrostatic free-surface evolution equation, is only first-order accurate for the time and space scales typically used in lake, estuarine, and coastal ocean studies.     

Numerical Solution of Fractional Advection-Dispersion Equation

Numerical schemes and stability criteria are developed for solution of the one-dimensional fractional advection-dispersion equation (FRADE) derived by revising Fick’s first law. Employing 74 sets of dye test data measured on natural streams, it is found that the fractional order F of the partial differential operator acting on the dispersion term varies around the most frequently occurring value of F = 1.65 in the range of 1.4 to 2.0. Two series expansions are proposed for approximation of the limit definitions of fractional derivatives. On this ground, two three-term finite-difference schemes?“1.3 Backward Scheme” having the first-order accuracy and “F.3 Central Scheme” possessing the F-th order accuracy?are presented for fractional order derivatives. The F.3 scheme is found to perform better than does the 1.3 scheme in terms of error and stability analyses and is thus recommended for numerical solution of FRADE. The fractional dispersion model characterized by the FRADE and the F.3 scheme can accurately simulate the long-tailed dispersion processes in natural rivers.     

Depth-Averaged Simulation of Supercritical Flow in Channel with Wavy Sidewall

Supercritical flow in a channel with a wavy sidewall is numerically simulated by solving the two-dimensional (2D) depth-averaged equations using two different second-order accurate finite-difference schemes: ADI and MAC. ADI is an implicit model that uses an alternating-direction-implicit (ADI) scheme to solve the governing equations. MAC is an explicit model employing the MacCormack two-step predictor-corrector scheme. To accurately simulate the wavy sidewall, both models solve the governing equations in transformed computational coordinates. Bottom friction is computed using the Manning formula and the effective stresses are modeled with a constant eddy-viscosity turbulence model. As is customary, the stresses due to depth-averaging are neglected. The computed water depth in the channel is compared with experimental data obtained by Mizumura. The effect of bottom friction, effective stresses, artificial viscosity, grid geometry, boundary conditions, and the Courant-Friedrichs-Lewy (CFL) number are investigated. Similarities and differences in the behavior of the models are observed and discussed.     

Synthesis of algorithms of control over the process of decomposition of nickel tetracarbonyl

Construction of an automated control system over the process of decomposition of nickel tetracarbonyl is considered. Confidence intervals of the obtained first-order and second-order models are analyzed. A model is described that allows one to compensate for the influence of perturbing factors and mutual influence of control effects on neighboring regions. Based on the algorithm presented, it is possible to construct an adaptive system of automated control over the process of decomposition of nickel tetracarbonyl.     

Evaluation of Advective Schemes for Estuarine Salinity Simulations

Several advective transport schemes are considered in the context of two-dimensional scalar transport. To review the properties of these transport schemes, results are presented for simple advective test cases. Wide variation in accuracy and computational cost is found. The schemes are then applied to simulate salinity fields in South San Francisco Bay using a depth-averaged approach. Our evaluation of the schemes in the salinity simulation leads to some different conclusions than those for the simple test cases. First, testing of a stable, but nonconservative Eulerian-Lagrangian scheme does not produce accurate results, showing the importance of mass conservation. Second, the conservative schemes that are stable in the simulation reproduce salinity data accurately independent of the order of accuracy of each scheme. Third, the leapfrog-central scheme was stable for the model problems but not stable in the unsteady, free surface computations. Thus, for the simulation of salinity in a strongly dispersive setting, the most important properties of a scalar advection scheme are stability and mass conservation.     

High-Order Compact Difference Scheme for Convection–Diffusion Problems on Nonuniform Grids

In this study, a high-order compact (HOC) scheme for solving the convection–diffusion equation (CDE) under a nonuniform grid setting is developed. To eliminate the difficulty in dealing with convection terms through traditional numerical methods, an upwind function is provided to turn the steady CDE into its equivalent diffusion equation (DE). After obtaining the HOC scheme for this DE through an extension of the optimal difference method to a nonuniform grid, the corresponding HOC scheme for the steady CDE is derived through converse transformation. The proposed scheme is of the upwind feature related to the convection–diffusion phenomena, where the convective–diffusion flux in the upstream has larger contributions than that in the downstream. Such a feature can help eliminate nonphysical oscillations that may often occur when dealing with convection terms through traditional numerical methods. Two examples have been presented to test performance of the proposed scheme. Under the same grid settings, the proposed scheme can produce more accurate results than the upwind-difference, central-difference, and perturbational schemes. The proposed scheme is suitable for solving both convection- and the diffusion-dominated flow problems. In addition, it can be extended for solving unsteady CDE. It is also revealed that efforts in optimizing the grid configuration and allocation can help improve solution accuracy and efficiency. Consequently, with the proposed method, solutions under nonuniform grid settings would be more accurate than those under uniform manipulations, given the same number of grid points.