Development of an algorithm for calculating stable solutions of the Saint-Venant equation using an upwind implicit difference scheme

The problem of numerical determination of Lyapunov-stable (exponential stability) solutions of the Saint-Venant equations system has remained open until now. The authors of this paper previously proposed an implicit upwind difference splitting scheme, but its practical applicability was not indicated there. In this paper, the problem is solved successfully, namely, an algorithm for calculating Lyapunov-stable solutions of the Saint-Venant equations system is developed and implemented using an upwind implicit difference splitting scheme on the example of the Big Almaty Canal (hereinafter BAC). As a result of the proposed algorithm application, it was established that: 1) we were able to perform a computational calculation of the numerical determination problem of the water level and velocity on a part of the BAC (10,000 meters) located in the Almaty region; 2) the numerical values of the water level height and horizontal velocity are consistent with the actual measurements of the parameters of the water flow in the BAC; 3) the proposed computational algorithm is stable; 4) the numerical stationary solution of the system of Saint-Venant equations on the example of the BAC is Lyapunov-stable (exponentially stable); 5) the obtained results (according to the BAC) show the efficiency of the developed algorithm based on an implicit upwind difference scheme according to the calculated time. Since we managed to increase the values of the difference grid time step up to 0.8 for calculating the numerical solution according to the proposed implicit scheme.

Well-posedness theory for nonlinear scalar conservation laws on networks

<p style='text-indent:20px;'>We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for scalar conservation laws on networks and establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case – for monotone fluxes with an upwind difference scheme – we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.</p>

Macroscopic Traffic Flow Characterization at Bottlenecks

Traffic congestion is a significant issue in urban areas. Realistic traffic flow models are crucial for understanding and mitigating congestion. Congestion occurs at bottlenecks where large changes in density occur. In this paper, a traffic flow model is proposed which characterizes traffic at the egress and ingress to bottlenecks. This model is based on driver response which includes driver reaction and traffic stimuli. Driver reaction is based on time headway and driver behavior which can be classified as sluggish, typical or aggressive. Traffic stimuli are affected by the transition width and changes in the equilibrium velocity distribution. The explicit upwind difference scheme is used to evaluate the Lighthill, Whitham, and Richards (LWR) and proposed models with a continuous injection of traffic into the system. A stability analysis of these models is given and both are evaluated over a road of length 10 km which has a bottleneck. The results obtained show that the behavior with the proposed model is more realistic than with the LWR model. This is because the LWR model cannot adequately characterize driver behavior during changes in traffic flow.

High-Accuracy Analysis Methods of Fluid Temperature Fluctuations at T-Junctions for Thermal Fatigue Evaluation

T-junctions are widely used for fluid mixing in nuclear power and process plants. Temperature fluctuations generated by the mixing of hot and cold fluids at a T-junction can cause thermal fatigue failure. The existing method for evaluating thermal fatigue leads to an over-conservative evaluation in part. CFD/FEM coupling analysis is a useful tool for the more rational evaluation of thermal fatigue. The present paper aims to verify another high-accuracy difference scheme used for simulating fluid temperature fluctuations at T-junctions for rational evaluation of thermal fatigue. In addition to the high-accuracy numerical methods presented in the previous paper, the TVD 2nd-order upwind difference scheme (2UD) is also used for the CFD analysis. In the present simulation, the TVD 2UD scheme is only applied to calculate the convective term of the energy equation to facilitate the comparison with the previous simulation. For comparison, the simulation conditions including the geometry of T-junction are the same as those in the experiments in the literature. The simulation results for distributions of time-averaged fluid temperature are very close to the experimental and previous simulation results. Especially, the predicted results of fluid temperature fluctuations of our concern also agree with the experimental and previous simulation results very well. The simulation results indicate that the TVD 2UD scheme is also a high-accuracy numerical method for the simulation of unsteady phenomena, along with the hybrid scheme.

A Finite-Volume Solution with a Bidirectional Upwind Difference Scheme for the Three-Dimensional Radiative Transfer Equation

A new calculation scheme is proposed for the explicitly discretized solution of the three-dimensional (3D) radiation transfer equation (RTE) for inhomogeneous atmospheres. To separate the independent variables involved in the 3D RTE approach, the spherical harmonic series expansion was used to discretize the terms, depending on the direction of the radiance, and the finite-volume method was applied to discretize the terms, depending on the spatial coordinates. A bidirectional upwind difference scheme, which is a specialized scheme for the discretization of the partial differential terms in the spherical harmonic-transformed RTE, was developed to make the equation determinate. The 3D RTE can be formulated as a simultaneous linear equation, which is expressed in the form of a vector–matrix equation with a sparse matrix. The successive overrelaxation method was applied to solve this equation. Radiative transfer calculations of the solar radiation in two-dimensional cloud models have shown that this method can properly simulate the radiation field in inhomogeneous clouds. A comparison of the results obtained using this method with those using the Monte Carlo method shows reasonable agreement for the upward flux, the total downward flux, and the intensities of radiance.