The modeling and numerical simulation of gas flow networks

Summary. A network formulation is introduced for the modeling and numerical simulation of complex gas transmission systems like a multi-cylinder internal combustion engine. Several simulation levels are discussed which result in different network representations of a specific system. Basic elements of a network are chambers of finite volume, straight pipes and connections like valves or nozzles. The pipe flow is modeled by the unsteady, one-dimensional Euler equations of gas dynamics. Semi-empirical approaches for the chambers and the connections yield differential-algebraic equations (DAEs) in time. The numerical solution is based on a TVD scheme for the pipe equations and a predictor-corrector method for the DAE-system. Simulation results for an internal combustion engine demonstrate the practical interest of the new approach. Received May 12, 1994 / Revised version received August 26, 1994     

An explicit third‐order numerical method for size‐structured population equations

A numerical method incorporating a combination of a difference scheme and several uniform and nonuniform quadrature rules is presented. The method is designed to solve size‐structured population equations with linear growth rate and nonlinear fertility and mortality rates. A detailed analysis of the global discretization error is carried out. Examples with known exact solutions have been solved numerically using the proposed method. The computations show that the global error is of third order as predicted by the theory. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 1–21, 2003     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

Numerical Algorithms - In this paper, we present an explicit six-step singularly P-stable Obrechkoff method of tenth algebraic order for solving second-order linear periodic and oscillatory initial...     

Exponential difference scheme for numerical simulation of viscous gas flow

Translated from Matematicheskie Modeli i Optimizatsiya Vychislitel'nykh Algoritmov, pp. 43–49, 1994.     

Modeling and simulation of a gas distribution pipeline network

This research study focuses on the modeling and simulation of a gas distribution pipeline network with a special emphasis on gas ducts. Gas ducts are the most important components of such kind of systems since they define the major dynamic characteristics. Isothermal, unidirectional flow is usually assumed when modeling the gas flow through a gas duct. This paper presents two simplified models derived from the set of partial differential equations governing the dynamics of the process. These models include the inclination term, neglected in most related papers. Moreover, two numerical schemes are presented for the integration of such models. Also, it is shown how the pressure drop along the pipe has a strong dependency with the inclination term. To solve the system dynamics through the proposed numerical schemes a based MATLAB-Simulink library was built. With this library it is possible to simulate the behavior of a gas distribution network from the individual simulation of each component. Finally, the library is tested through three application examples, and results are compared with the existing ones in the literature.     

An explicit second-order numerical scheme for mean-field forward backward stochastic differential equations

Numerical Algorithms - In this paper, we propose an explicit second-order scheme for solving decoupled mean-field forward backward stochastic differential equations. Its stability is theoretically...     

Variational inequality model for the simulation of gas transmission networks

The steady-state simulation of a gas transmission network involves the solution of a system of nonlinear algebraic equations subject to a set of inequality constraints. The purpose of this paper is to consider how the problem can be cast into a variational inequality framework using the dual extremum principles described by Noble and Sewell in 1972. The reliability of the model is tested on two networks supplied by the British Gas Corporation.     

An explicit second order spectral element method for acoustic waves

The acoustic wave equation is here discretized by conforming spectral elements in space and by the second order leap-frog method in time. For simplicity, homogeneous boundary conditions are considered. A stability analysis of the resulting method is presented, providing an upper bound for the allowed time step that is proportional to the size of the elements and inversely proportional to the square of their polynomial degree. A convergence analysis is also presented, showing that the convergence error decreases when the time step or the size of the elements decrease or when the polynomial degree increases. Several numerical results illustrating these results are presented. Mathematics subject classifications (2000) 65M06, 65M70, 65M12 This work was supported by MIUR (PRIN 2001 “Metodi numerici avanzati per equazioni alle derivate parziali di interesse applicativo”).     

Optimization methods for pipeline transportation of natural gas with variable specific gravity and compressibility

In this paper, the problem of flow maximization in pipeline systems for transmission of natural gas is addressed. We extend previously suggested models by incorporating the variation in pipeline flow capacities with gas specific gravity and compressibility. Flow capacities are modeled as functions of pressure, compressibility and specific gravity by the commonly-used Weymouth equation, and the California Natural Gas Association method is used to model compressibility as a function of specific gravity and pressure. The sources feeding the transmission network do not necessarily supply gas with equal specific gravity. In our model, it is assumed that when different flow streams enter a junction point, the specific gravity of the resulting flow is a weighted average of the specific gravities of entering flows. We also assume the temperature to be constant, and the system to be in steady state. Since the proposed model is non-convex, and global optimization hence can be time consuming, we also propose a heuristic method based on an iterative scheme in which a simpler NLP model is solved in each iteration. Computational experiments are conducted in order to assess the computability of the model by applying a global optimizer, and to evaluate the performance of the heuristic approach. When applied to a wide set of test instances, the heuristic method provides solutions with deviations less than 10% from optimality, and in many instances turns out to be exact. We also report several experiments demonstrating that letting the compressibility and the specific gravity be global constants can lead to significant errors in the estimates of the total network capacity.     

An explicit fourth-order staggered finite-difference time-domain method for Maxwell's equations

A new explicit fourth-order accurate staggered finite-difference time-domain (FDTD) scheme is proposed and applied to electromagnetic wave problems. It is fourth-order accurate in both space and time, conditionally stable, and highly efficient (with respect to Yee's scheme) and still retains much of the original simplicity of Yee's scheme. Both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented. Numerical examples are shown to illustrate the efficiency of the method.     

A numerical method for identifying the location of a fluid leak in a pipeline

On the basis of mathematical models for studying gas or liquid flows in long-distance pipelines, a numerical method is developed that allows us to determine the coordinates of the fluid leak points by measuring data for the flow parameters (pressure and flow rate) both in stationary and nonstationary cases. The method is a modification of the Newton method applied to minimizing the functional of the mean-square deviation of the calculated parameters from the measured data.     

Application of the particle-in-cell method for numerical simulation of sheath plasma

We give a survey of papers on the numerical simulation of the sheath plasma using the particle-in-cell method. We study the problem of the behavior of a plasma bounded in the longitudinal direction of an absorbing wall. The model studied contains charged particles (electrons and ions) that move subject to a self-consistent electrostatic field. New pairs of particles are generated in the region of a distributed source. As a numerical model we use the electrostatic “particle-in-cell” method supplemented with the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using a Monte Carlo method. Only electron-ion collisions are taken into account. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 101–131.     

An efficient method for nonlinearly constrained networks

Many nonlinear network flow problems (in addition to the balance constraints in the nodes and capacity constraints on the arc flows) have nonlinear side constraints, which specify a flow relationship between several of the arcs in the network flow model. The short-term hydrothermal coordination of electric power generation is an example of this type. In this work we solve this kind of problem using an approach in which the efficiency of the well-known techniques for network flow can be preserved. It lies in relaxing the side constraints in an augmented Lagrangian function, and minimizing a sequence of these functions subject only to the network constraints for different estimates of the Lagrange multipliers of the side constraints. This method gives rise to an algorithm, which combines first- and superlinear-order multiplier methods to estimate these multipliers. When the number of free variables is very high we can obtain a superlinear-order estimate by means of the limited memory BFGS method fitted to our problem. An extensive computational comparison with other methods has been performed. The numerical results reported indicate that the algorithm described may be employed advantageously to solve large-scale network flow problems with nonlinear side constraints.     

A low-Mach number method for the numerical simulation of complex flows

A new numerical procedure which considers a modification to the artificial acoustic stiffness correction method (AASCM) is here presented, to perform simulations of low Mach number flows with the compressible Navier–Stokes equations. An extra term is added to the energy fluxes instead of using an energy source correction term as in the original model. This new scheme re-scales the speed of sound to values similar to the flow velocity, enabling the use of larger time steps and leading to a more stable numerical method. The new method is validated performing Large Eddy Simulations on test problems. The effect of a crucial numerical parameter alpha is evaluated as well as the robustness of the method to variations of the Mach number. Numerical results are compared to the existing experimental data showing that the new method achieves good agreement increasing the time-step, and therefore accelerating the computation for low-Mach convective flows.     

A modified numerical manifold method for simulation of finite deformation problem

With many people contributing to its modifications and advancements, the numerical manifold method (NMM) is now recognized as an efficient tool to solve the continuum–discontinuum coupling problem in geotechnical engineering. However, false solutions have been found when modeling finite deformation problems using the original NMM. Based on the finite deformation theory, a modified version of NMM is derived from the weak form of conservation of momentum and the corresponding traction boundary condition. By taking the dual cover system as the displacement approximation, the governing equations of the modified NMM are formulated. A comparison of the governing equations of the original NMM and modified NMM illustrates the reason that the original NMM is not suitable for simulation of finite deformation problems. Three numerical examples are investigated to verify the capability of proposed method to predict static and dynamic finite deformation response. Numerical results show that the modified NMM eliminates the errors caused by large rotation and large strain, and obtains a good agreement with analytical solutions and the finite element method.     

An efficient numerical method for singular perturbation problems

In this paper, we propose a method for the numerical solution of singularly perturbed two-point boundary-value problems (BVPs). First, we develop two schemes to integrate initial–value problem (IVP) for system of two first-order differential equations, and then by using these schemes we solve the BVP. Precisely, we convert the second-order BVP into a system of first-order differential equations, and then apply the numerical schemes to obtain the solution. In order to get an initial condition for the system, we use the asymptotic approximate solution. Error estimates are derived and numerical examples are provided to illustrate the present method.     

An averaging scheme for macroscopic numerical simulation of nonconvex minimization problems

Averaging or gradient recovery techniques, which are a popular tool for improved convergence or superconvergence of finite element methods in elliptic partial differential equations, have not been recommended for nonconvex minimization problems as the energy minimization process enforces finer and finer oscillations and hence at the first glance, a smoothing step appears even counterproductive. For macroscopic quantities such as the stress field, however, this counterargument is no longer true. In fact, this paper advertises an averaging technique for a surprisingly improved convergence behavior for nonconvex minimization problems. Similar to a finite volume scheme, numerical experiments on a double-well benchmark example provide empirical evidence of superconvergence phenomena in macroscopic numerical simulations of oscillating microstructures. AMS subject classification (2000)  65K10,65N30