Conservation Law Model Research Articles

A filtered Chebyshev spectral method for conservation laws on network

We propose a spectral method based on the implementation of Chebyshev polynomials to study a model of conservation laws on network.We avoid the Gibbs phenomenon near shock discontinuities by implementing an exponential filter in the frequency space coupled with the Midpoint method for time marching. This strategy consists in adding a local viscosity into the model, and is suitable to contrast the spurious oscillations appearing in the profile of the solution. We prove the convergence of the semi-discrete method by using the compensated compactness theorem and exploiting the regularity of viscous profiles. Thanks to several simulations, we make a comparison between the implementation of the proposed method with a first order finite volume scheme.

Prediction of ice accretion and aerodynamic performance analysis of NACA 2412 aerofoil

AbstractAdverse meteorological conditions often contribute to the formation of ice on aircraft wing section, engine nacelle and other parts leading to the loss of lift coefficient and increase in drag coefficient affecting aircraft control and stability. This paper addresses the problem of in-flight icing on an asymmetric aerofoil under three different ambient and cloud conditions. The study involves prediction of the leading-edge ice thickness using a numerical model developed from the mass and energy conservation law and Messinger freezing fraction model at the same Reynolds number. Later on, degradation in the aerodynamic performance of the iced aerofoil was also investigated using the computational fluid dynamics (CFD) technique, taking the flow field around a 2D aerofoil geometry into account. The aerodynamic study indicates that cumulus clouds embedded with stratified clouds contribute to the formation of mixed ice on aerofoil leading edge and causes the worst icing scenario reducing the lift coefficient to 90% and increasing the drag coefficient to 800% for the same ambient conditions.

Exact solution of the flux perturbed Riemann problem for Cargo-LeRoux model in a van der Waals gas

In this paper, we consider the pressureless Cargo-LeRoux model of conservation laws represented by a system of quasi-linear partial differential equations derived from the one-dimensional Euler equations with constant gravity. Considering flux perturbation of a van der Waals gas equation of state, we derive the exact solution of this Riemann problem based on the elementary wave analysis including shock wave, rarefaction wave and contact discontinuity wave. The Newton-Raphson method of two variables is applied to find the densities across the contact discontinuity wave of two nonlinear algebraic equations in all possible combinations of waves. Finally, this study exclusively reveals the influence of the growth of van der Waals excluded volume on the physical quantities: density, potential, velocity and total pressure through a series of test cases.

Interaction of delta shock waves for a nonsymmetric Keyfitz–Kranzer system of conservation laws

In this work, the mechanism for the formation of the delta shock wave is analyzed to deal with interaction of delta shock waves and contact discontinuities for a system of Keyfitz–Kranzer type by means of analysis and solutions of Riemann problems. A set of numerical experiments are provided, illustrating the theoretical findings numerically. A brief survey of the Keyfitz–Kranzer systems as a base model of fundamental nonlinear phenomena in applications is provided aiming to shed light on the intricate wave structure for other related models of conservation laws appearing in applied sciences.

Entropy admissibility of the limit solution for a nonlocal model of traffic flow

We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $\rho$ ahead. The averaging kernel is of exponential type: $w_\varepsilon (s)=\varepsilon^{-1} e^{-s / \varepsilon}$. For any decreasing velocity function $v$, we prove that, as $\varepsilon \to 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law $\rho_t + {(\rho v(\rho))}_x = 0$.

Entropy stable reduced order modeling of nonlinear conservation laws

Reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model. We introduce projection-based hyper-reduced models of nonlinear conservation laws which are globally conservative and inherit a semi-discrete entropy inequality independently of the choice of basis and choice of parameters.

On Traffic Flow with Nonlocal Flux: A Relaxation Representation

We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $\rho$ ahead. The averaging kernel is of exponential type: $w_\varepsilon(s)=\varepsilon ^{-1} e^{-s/\varepsilon}$. By a transformation of coordinates, the problem can be reformulated as a $2\times 2$ hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter $\varepsilon $. Letting $\varepsilon\to 0$, the limit yields a weak solution to the corresponding conservation law $\rho_t + ( \rho v(\rho))_x=0$. In the case where the velocity $v(\rho)= a-b\rho$ is affine, using the Hardy-Littlewood rearrangement inequality we prove that the limit is the unique entropy-admissible solution to the scalar conservation law.

High-Resolution Schemes for Stochastic Nonlinear Conservation Laws

In this paper, high-resolution methods for stochastic Buckley Leverett equation and stochastic polymer flooding system are used. The models are augmented with random initial conditions. The numerical solutions are obtained to assess the performance of the methods on the stochastic models of conservation laws for different flow situations. The mean and standard deviation of the stochastic solutions are evaluated. It is investigated that, to what extent recent, the randomness in initial data can affect and be useful in this framework. The consistency, stability and the local truncation error of the methods are proved. Numerical experiments with different scenarios simulate the saturation profile of the models and demonstrating the accuracy of the schemes.

An upwind local radial basis functions-differential quadrature (RBFs-DQ) technique to simulate some models arising in water sciences

The main aim of the current paper is to propose an efficient numerical procedure based on the meshless method for solving some models of conservation laws. In the current investigation, we combine the upwind local radial basis function-differential quadrature (LRBF-DQ) method with the space-splitting idea for solving some conservation laws equations. The upwind LRBF-DQ is applied to discrete the spatial direction as this procedure yields a nonlinear system of ordinary differential equations (ODEs). Then, we solve the achieved system of ODEs by using the exponential time differencing (ETD) method. The accuracy and stability of the developed technique are shown for some conservation laws equations.

Riemann problem for the pressureless Cargo–LeRoux model by flux perturbation

By introducing a potential, a new formulation model of conservation laws was derived by Cargo and LeRoux from the Euler equations with a constant gravity (Cargo and LeRoux in C R Acad Sci Paris Serie I Math 318:73–76, 1994). It gives a conservative form extension of the Euler equations without external forces. This paper is concerned with the pressureless flow of the Cargo–LeRoux model. By using a flux perturbation, the solutions with three kinds of structures of the Riemann problem are constructed. The new type of structure simultaneously including delta-shock and vacuum state appears in solution. The Riemann solutions also give an interesting extension of those to the pressureless Euler equations without external forces.

The piecewise parabolic method for elastic-plastic flow in solids

A numerical technique of high-order piecewise parabolic method in combination of HLLD (”D” denotes Discontinuities) Riemann solver is developed for the numerical simulation of elastic-plastic flow. The introduction of the plastic effect is realized by decomposing the total deformation gradient tensor as the product of elastic and plastic deformation gradient tensors and adding plastic source term to the conservation law model equation with the variable of the elastic deformation gradient tensor. For the solution of the resulting inhomogeneous equation system, a temporal splitting strategy is adopted and a semi-implicit scheme is performed to solve the ODES in the plastic step, which is conducted to account for the contributions from plastic source terms. As seen from the results of test cases involving large deformation and high strain rate, the computational model used can reflect the characteristics of constitutive relation of material under strong impact action and our numerical method can realize the exact simulation of the elastic-plastic behavior of solid material, especially the accurate capture of the elastic-plastic waves. Further, it could also deal with high-speed impact problems with multi-material components, catching material interfaces correctly and keeping the interfaces sharp, when combined with interface tracking technique such as the level-set algorithm.

Probabilistic Representations and Numerical Algorithms for Classical and Viscosity Solutions of the Cauchy Problem for Quasilinear Parabolic Systems

We propose two approaches which allow us to construct probabilistic representations of classical and viscosity solutions of the Cauchy problem for a system of quasilinear parabolic equations. Based on these representations, we develop two numerical algorithms to construct the required solution. The system under consideration arises as a mathematical model of parabolic conservation laws. Bibliography: 14 titles.

A mechanistic modeling framework for gas‐phase adsorption kinetics and fixed‐bed transport

Adsorption is a complex physicochemical process involving interparticle transport, interphase mass‐transfer, intraparticle diffusion, and surface reactions. Although the exact description of the adsorption process will inevitably vary from system to system, it will always be governed by those primary mechanisms. Therefore, by devising a model framework that can inherently include those mechanisms, it would be possible to create a modeling platform on which many different adsorption problems could be solved numerically. To accomplish this task, a generalized 1‐D conservation law model was created to include the necessary mechanisms of adsorption on several different geometrical domains. Specific model applications for adsorption were developed under that framework and validated using experimental data available in literature or obtained in this work. This modeling platform makes it easier to model various adsorption problems and develop new adsorption models because of the common treatment of the mathematics governing the physical processes. © 2017 American Institute of Chemical Engineers AIChE J, 2017

A numerical study on a nonlinear conservation law model pertaining to manufacturing system

This paper deals with a numerical scheme applied to a conservation law model of manufacturing system incorporating yield loss. Yield loss involving a singular term has been considered. Even though an explicit form of the material density in a production system can be obtained under certain assumptions, in general, it is difficult to get an explicit form of the material density. On the other hand, the singular term in a conservation law model imposes severe challenges for the numerical approximations on regular grids. Moreover, an approximate solution often converges to a wrong weak solution. A finite volume type numerical scheme has been studied. The convergence of the numerical solution towards entropy solution (in the Kruzkov sense) is proved. Numerical experiments are presented to get the overview of density distribution and outflux of the system.

A conservation law model for bidensity suspensions on an incline

We study bidensity suspensions of a viscous fluid on an incline. The particles migrate within the fluid due to a combination of gravity-induced settling and shear induced migration. We propose an extension of a recent model (Murisic et al., 2013) for monodisperse suspensions to two species of particles, resulting in a hyperbolic system of three conservation laws for the height and particle concentrations. We analyze the Riemann problem and show that the system exhibits three-shock solutions representing distinct fronts of particles and liquid traveling at different speeds as well as singular shock solutions for sufficiently large concentrations, for which the mechanism is essentially the same as the single-species case. We also consider initial conditions describing a fixed volume of fluid, where solutions are rarefaction–shock pairs, and present a comparison to recent experimental results. The long-time behavior of solutions is identified for settled mono- and bidisperse suspensions and some leading-order asymptotics are derived in the single-species case for moderate concentrations.

Parameter extraction of complex production systems via a kinetic approach

Continuum models of re-entrant production systems are developed that treat the flow of products in analogy to traffic flow. Specifically, the dynamics of material flow through a re-entrant factory via a parabolic conservation law is modeled describing the product density and flux in the factory. The basic idea underlying the approach is to obtain transport coefficients for fluid dynamic models in a multi-scale setting simultaneously from Monte Carlo simulations and actual observations of the physical system, i.e. the factory. Since partial differential equation (PDE) conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, a re-entrant manufacturing system is modeled using a diffusive PDE. The specifics of the production process enter into the velocity and diffusion coefficients of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations. With the traffic flow-like PDE model, the transient behavior of the discrete event simulation (DES) model according to the averaged influx, which is obtained out of discrete event experiments, is predicted. The work brings out an almost universally applicable tool to provide rough estimates of the behavior of complex production systems in non-equilibrium regimes.

Continuous-time link-based kinematic wave model: formulation, solution existence, and well-posedness

We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traffic networks based on the scalar conservation law model. Derivation of the LKWM involves the variational principle for the Hamilton–Jacobi equation and junction models defined via the notions of demand and supply. We show that the proposed LKWM can be formulated as a system of differential algebraic equations (DAEs), which captures shock formation and propagation, as well as queue spillback. The DAE system, as we show in this paper, is the continuous-time counterpart of the link transmission model. In addition, we present a solution existence theory for the continuous-time network model and investigate continuous dependence of the solution on the initial data, a property known as well-posedness. We test the DAE system extensively on several small and large networks and demonstrate its numerical efficiency.

Nonlinear conservation law model for production network considering yield loss

A mathematical model describing yield loss in a production network has been introduced. Mathematical properties of the continuum model are discussed. Existence, uniqueness and stability of the solution are demonstrated through weak formulation and entropy criteria. Front tracking method is implemented to construct approximate solutions. Estimates of the solutions are also provided.

Existence of optima and equilibria for traffic flow on networks

This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.

Nonlinear adaptive PID control for greenhouse environment based on RBF network.

This paper presents a hybrid control strategy, combining Radial Basis Function (RBF) network with conventional proportional, integral, and derivative (PID) controllers, for the greenhouse climate control. A model of nonlinear conservation laws of enthalpy and matter between numerous system variables affecting the greenhouse climate is formulated. RBF network is used to tune and identify all PID gain parameters online and adaptively. The presented Neuro-PID control scheme is validated through simulations of set-point tracking and disturbance rejection. We compare the proposed adaptive online tuning method with the offline tuning scheme that employs Genetic Algorithm (GA) to search the optimal gain parameters. The results show that the proposed strategy has good adaptability, strong robustness and real-time performance while achieving satisfactory control performance for the complex and nonlinear greenhouse climate control system, and it may provide a valuable reference to formulate environmental control strategies for actual application in greenhouse production.