Lax-Wendroff Scheme Research Articles

AN IMPROVED MODEL FOR REDUCED-ORDER PHYSIOLOGICAL FLUID FLOWS

An improved one-dimensional mathematical model based on the Pulsed Flow Equations (PFE) is derived by integrating the axial component of the momentum equation over the transient Womersley velocity profile, providing a dynamic momentum equation whose coefficients are smoothly varying functions of the spatial variable. The resulting momentum equation along with the continuity equation and pressure-area relation form our reduced-order model for physiological fluid flows in one dimension and are aimed at providing accurate and fast-to-compute global models for physiological systems represented as networks of quasi one-dimensional fluid flows. The consequent nonlinear coupled system of equations is solved by the Lax-Wendroff scheme and is then applied to an open model arterial network of the human vascular system containing the largest 55 arteries. The proposed model with functional coefficients is compared with current classical one-dimensional theories which assume steady state Hagen-Poiseuille velocity profiles, either parabolic or plug-like, throughout the whole arterial tree. The effects of the nonlinear term in the momentum equation and different strategies for bifurcation points in the network, as well as the various lumped parameter outflow boundary conditions for distal terminal points are also analyzed. The results show that the proposed model can be used as an efficient tool for investigating the dynamics of reduced-order models of flows in physiological systems and would, in particular, be a good candidate for the one-dimensional, system-level component of geometric multiscale models of physiological systems.

THEORETICAL INVESTIGATION INTO A PRESSURE WAVE SUPERCHARGER

Pressure wave supercharger (PWS) is widely used in different applications especially for charging the internal combustion engines. This device utilizes the pressure waves issued the exhaust manifold. These waves transmitted from the exhaust gas side to the air side causing compression effect required for charging process. The present work aims to build a mathematical model to address the behavior of PWS at a wide range of operating conditions. The proposed model employs the basic conservation equations of continuity, momentum and energy as well as the species transportation. The gas flow is treated as 1-D, time dependent, and non-reactive compressible fluid flow. These equations are solved together numerically by using two steps Lax-Wendroff scheme. This technique enables to simulate the pressure waves more accurately and precisely. A computer code has been built to simulate the effect of many parameters on PWS performance. These parameters are dimensions and cells size, number of cells, rotational speed of PWS, engine speed and the exhaust gas pressure. Synchronization between the engine speed and PWS speed requires mathematical coupling between the engine cycle and PWS. Therefore, real cycle simulation is performed taking into consideration combustion processes, valve timing, and the amount of residual gases. The results of the present model are compared with another data to validate the model. The comparison shows fair agreement. Using PWS with internal combustion engines leads to enhancing the engine power, volumetric efficiency and reducing NOx emissions.

Numerical solution of compressible steady flows in a 2D GAMM channel and DCA 18% profile

• Abstract: The article presents results of a numerical solution of subsonic and transonic flows described by the system of Euler equations in 2D flows in a channel and around a profile. Authors used Lax-Wendroff scheme to numerically solve the flows in a GAMM channel and around half DCA 18% profile. Authors programmed the mesh generator of the type C for profile with a blunt leading edge.

The early influence of peter lax on computational hydrodynamics and an application of lax-friedrichs and lax-wendroff on triangular grids in lagrangian coordinates

We give a brief discussion of some of the contributions of Peter Lax to Computational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We develop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax-Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh's infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.

Run-Up of Solitary Waves on Twin Conical Islands Using a Boussinesq Model

This paper investigates the interaction of solitary waves (representative of tsunamis) with idealized flat-topped conical islands. The investigation is based on simulations produced by a numerical model that solves the two-dimensional Boussinesq-type equations of Madsen and Sørensen using a total variation diminishing Lax–Wendroff scheme. After verification against published laboratory data on solitary wave run-up at a single island, the numerical model is applied to study the maximum run-up at a pair of identical conical islands located at different spacings apart for various angles of wave attack. The predicted results indicate that the maximum run-up can be attenuated or enhanced according to the position of the second island because of wave refraction, diffraction, and reflection. It is also observed that the local wave height and hence run-up can be amplified at certain gap spacing between the islands, owing to the interference between the incident waves and the reflected waves between islands.

One Dimensional Modeling and Experimental Validation of Single Cylinder Pneumatic Combustion Hybrid Engine

The objective of this paper is to present and to validate a numerical model of a single-cylinder pneumatic-combustion hybrid engine. The model presented in this paper contains 0-D sub-models for non-spatially distributed components: Engine cylinder, Air tank, wall heat losses. 1-D sub-models for spatially distributed components are applied on the compressive gas flows in pipes (intake, exhaust and charging). Each pipe is discretized, using the Two-Steps Lax-Wendroff scheme (LW2) including Davis T.V.D. The boundaries conditions used at pipe ends are Method Of Characteristics (MOC) based. In the specific case of a valve, an original intermediate volume MOC based boundary condition is used. The numerical results provided by the engine model are compared with the experimental data obtained from a single cylinder prototype hybrid engine on a test bench operating in 4-stroke pneumatic pump and 4 stroke pneumatic motor modes. In each mode, the prediction of the mass flow rates, amplitude and timing of the charging pipe waves are satisfactory, without using any discharge coefficient. Indicated work and p-V diagrams are similar between simulation and measurements in the case of pneumatic pump mode. For the pneumatic motor mode the model underestimates cylinder pressure during the charging process.

Investigation of gas dynamics numerical schemes based on the solution of one-dimensional test problems

The application of the optimized dispersion relation preserving scheme (DRP-scheme) in combination with the explicit optimized two-layer Runge-Kutta scheme is presented to solve a system of one-dimensional and quasi-one-dimensional Euler equations using as an example the solution of four test problems, namely, discontinuity disintegration in a tube (Sod’s problem); transfer of the lowamplitude Gaussian pulse; acoustic wave propagation through the transonic nozzle; acoustic wave-shock interaction. Also given are the comparison of the calculation results using different schemes: DRP, CABARET, CE-SE and the standard Lax-Wendroff schemes as well as the solutions obtained with the use of software packages.

A Maxwell–Schrödinger solver for quantum optical few-level systems

The msprop program presented in this work is capable of solving the Maxwell–Schrödinger equations for one or several laser fields propagating through a medium of quantum optical few-level systems in one spatial dimension and in time. In particular, it allows to numerically treat systems in which a laser field interacts with the medium with both its electric and magnetic component at the same time. The internal dynamics of the few-level system is modeled by a quantum optical master equation which includes coherent processes due to optical transitions driven by the laser fields as well as incoherent processes due to decay and dephasing. The propagation dynamics of the laser fields is treated in slowly varying envelope approximation resulting in a first order wave equation for each laser field envelope function. The program employs an Adams predictor formula second order in time to integrate the quantum optical master equation and a Lax–Wendroff scheme second order in space and time to evolve the wave equations for the fields. The source function in the Lax–Wendroff scheme is specifically adapted to allow taking into account the simultaneous coupling of a laser field to the polarization and the magnetization of the medium. To reduce execution time, a customized data structure is implemented and explained. In three examples the features of the program are demonstrated and the treatment of a system with a phase-dependent cross coupling of the electric and magnetic field component of a laser field is shown. Program summary Program title: msprop Catalogue identifier: AEHR_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHR_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 507 625 No. of bytes in distributed program, including test data, etc.: 10 698 552 Distribution format: tar.gz Programming language: C (C99 standard), Mathematica, bash script, gnuplot script Computer: Tested on x86 architecture Operating system: Unix/Linux environment RAM: Less than 30 MB Classification: 2.5 External routines: Standard C math library, accompanying bash script uses gnuplot, bc (basic calculator), and convert (ImageMagick) Nature of problem: We consider a system of quantum optical few-level atoms exposed to several near-resonant continuous-wave or pulsed laser fields. The complexity of the problem arises from the combination of the coherent and incoherent time evolution of the atoms and its dependence on the spatially varying fields. In systems with a coupling to the electric and magnetic field component the simultaneous treatment of both field components poses an additional challenge. Studying the system dynamics requires solving the quantum optical master equation coupled to the wave equations governing the spatio-temporal dynamics of the fields [1,2]. Solution method: We numerically integrate the equations of motion using a second order Adams predictor method for the time evolution of the atomic density matrix and a second order Lax–Wendroff scheme for iterating the fields in space [3]. For the Lax–Wendroff scheme, the source function is adapted such that a simultaneous coupling to the polarization and the magnetization of the medium can be taken into account. Restrictions: The evolution of the fields is treated in slowly varying envelope approximation [2] such that variations of the fields in space and time must be on a scale larger than the wavelength and the optical cycle. Propagation is restricted to the forward direction and to one dimension. Concerning the description of the atomic system, only a finite number of basis states can be treated and the laser-driven transitions have to be near-resonant such that the rotating-wave approximation can be applied [2]. Unusual features: The program allows the dipole interaction of both the electric and the magnetic component of a laser field to be taken into account at the same time. Thus, a system with a phase-dependent cross coupling of electric and magnetic field component can be treated (see Section 4.2 and [4]). Concerning the implementation of the data structure, it has been optimized for faster memory access. Compared to using standard memory allocation methods, shorter run times are achieved (see Section 3.2). Additional comments: Three examples are given. They each include a readme file, a Mathematica notebook to generate the C-code form of the quantum optical master equation, a parameter file, a bash script which runs the program and converts the numerical data into a movie, two gnuplot scripts, and all files that are produced by running the bash script. Running time: For the first two examples the running time is less than a minute, the third example takes about 12 minutes. On a Pentium 4 (3 GHz) system, a rough estimate can be made with a value of 1 second per million grid points and per field variable.

Note on a 2DH finite element model of tidal flow of the North Passage of the partially-mixed Changjiang River estuary, China

A two-dimensional horizontal (2DH) numerical model of flow is established by applying the Galerkin finite element method (FEM). The software EasyMesh is used to triangulate the modeled planar domain. The two-step Lax–Wendroff scheme is used for integrating the equations in order to avoid the nonlinear iterative calculation. The model is validated through numerical experiments on flow in an idealized rectangular flume. The model is then used to study tidal flow within the North Passage of the partially-mixed Changjiang River estuary. The dry–wet method is adopted to deal with the moving boundary hydrodynamic problem. Comparisons are made between modeled and measured hydrographic data for two stations. Modeled results are shown to be in reasonable agreement with measured ones. Horizontal distributions of modeled tidal flow velocity vectors within the North Passage are obtained during the maximum flood tide, the high slack water, the maximum ebb tide, and the low slack water, respectively. This two-dimensional horizontal finite element model was found to be well suited to the complexities of the North Passage of the partially-mixed Changjiang River estuary.

A flux-limiter method for modeling blood flow in the aorta artery

In this paper, we extend a flux-limiter method to obtain a 1D haemodynamic model of the aorta artery. For the validation of the proposed method, we use experimental data obtained from a fluid in a latex tube. We compare the results obtained by using the extended flux-limiter method, the Richtmyer two-step Lax–Wendroff scheme and a modified Steger–Warming method. The flux-limiter method provides accurate results avoiding numerical oscillation and numerical dispersion in the approximated solution. Finally, we simulate the fluid–structure interaction of the aorta artery with the flux-limiter solver.

Eliminating Discharge Imbalances in Predicting Shallow Water Flows with Shocks

The shallow water equations are widely used in modelling environmental flows. Being a hyperbolic system of differential equations, they admit shocks that represent hydraulic jumps and bores. Although the water surface can be solved satisfactorily with the modern shock-capturing schemes, the predicted flow rate often suffers from imbalances where shocks occur, eg the mass conservation is violated by failing to maintain a constant discharge rate at every cross-section in a steady open channel flow. A total-variation-diminishing Lax-Wendroff scheme is developed, and used to demonstrate how to achieve an exact flux balance. The performance of the proposed methods is inspected through some test cases, which include 1- and 2-dimensional, flat and irregular bed scenarios. The proposed methods are shown to preserve the mass exactly, and can be easily extended to other shock-capturing models.

On some fast well-balanced first order solvers for nonconservative systems

The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good well-balanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.

Sensitivity analysis of lossy nonuniform multiconductor transmission lines based on the Lax‐Wendroff technique

AbstractThis paper applies the Lax‐Wendroff technique, usually used in fluid dynamics, to transmission line sensitivity analysis. A second‐order‐accurate Lax‐Wendroff difference scheme for sensitivity analysis of both uniform and nonuniform transmission lines is derived. Based on this scheme, a new method for analyzing multiconductor transmission line sensitivity, which does not need to be decoupled, is presented by combining with matrix operations. Using numerical experiments, the proposed method is compared with the characteristic method and the fast Fourier transform approach. With the presented method, the sensitivity of a nonlinear circuit including nonuniform multiconductor transmission lines is analyzed and the results are verified by the HSPICE perturbation method. The proposed method can be applied to either linear or nonlinear circuits, which include lossy nonuniform multiconductor transmission lines and is proved to be efficient. Copyright © 2009 John Wiley & Sons, Ltd.

A numerical model characterizing internal gravity wave propagation into the upper atmosphere

A new two-dimensional, time-dependent and fully nonlinear model is developed to numerically simulate plane wave motions for internal gravity waves in a non-isothermal and windy atmosphere that accounts for the dissipation due to eddy and molecular processes. The atmosphere has been treated as a well mixed total gas with a constant mean molecular weight. The effects of Rayleigh friction and Newtonian cooling are applied near the upper boundary of the model to simulate the radiation conditions as well as to act as a sponge layer; lateral boundaries are periodic over a horizontal wavelength to simulate a horizontally infinite domain. The thermal excitation to initiate upward propagating waves is spatially localized in the troposphere and is a Gaussian function of time. A time-splitting technique is applied to the finite difference equations that are derived from the Navier–Stokes equations. The time integration for these highly coupled equations is performed using an explicit second order Lax–Wendroff scheme and an implicit Newton–Raphson scheme. The wave solutions derived from the model are found to be broadly agreeable with those derived from a Wentzel–Kramers–Brillouin theory.

Numerical modeling of two-phase transonic flow

Our work is aimed at the development of numerical method for the modeling of transonic flow of wet steam including condensation/evaporation phase change. We solve a system of PDE’s consisting of Euler or Navier-Stokes equations for the mixture of vapor and liquid droplets and transport equations for the integral parameters describing the droplet size spectra. Numerical method is based on a fractional step technique due to the stiff character of source terms, i.e. we solve separately the set of homogenous PDE’s by the finite volume method and the remaining set of ODE’s either by explicit Runge-Kutta or implicit Euler method. The finite volume method is based on the Lax-Wendroff scheme with conservative artificial dissipation terms for structured grid. We also note result achieved by recently developed finite volume method with VFFC scheme. We discuss numerical results of steady and unsteady two-phase transonic flow in 2D nozzle, 2D and 3D turbine cascade and 2D turbine stage with moving rotor cascade.

Numerical Solution of a Nonlocal Problem Modelling Ohmic Heating of Foods

Abstract An upwind and a Lax-Wendroff scheme are introduced for the solution of a one-dimensional non-local problem modelling ohmic heating of foods. The schemes are studied regarding their consistency, stability, and the rate of convergence for the cases that the problem attains a global solution in time. A high resolution scheme is also introduced and it is shown that it is total-variation-stable. Finally some numerical experiments are presented in support of the theoretical results.

Three Dimensional MHD Simulationof Comet Hale-Bopp Tail

The interaction between comet Hale-Bopp tail with the solar wind is investigated in the present paper using magneto-hydrodynamic (MHD) numerical simulation, which accounts for the presence of the interplanetary magnetic field (IMF). The simulation is based on three-dimensional Lax-Wendroff explicit scheme, providing second-order accuracy in space and time. The ions produced from the nucleus of the comet will add considerable effects on the microstructure of the solar wind, thus severely altering its physical properties. The present simulation focuses on careful analysis of these properties by means of simulating the behavior of the comet Hale-Bopp’s tail at 1 AU from the sun. These properties include the changes of the plasma density, particle velocity, IMF changes, pressure, and internal energy. The results indicated that comet tail will highly disturb the physical properties of the solar wind for a considerable distance. These changes reflect the effect of including the source term in the present simulation. It is shown that the comet tail will retain its original shape faster as it approaches the sun. Practical comparisons are also presented in the present research with earlier work. The present simulation was made using MATLAB program.

Three Dimensional MHD Simulationof Comet Hale-Bopp Tail

The interaction between comet Hale-Bopp tail with the solar wind is investigated in the present paper using magneto-hydrodynamic (MHD) numerical simulation, which accounts for the presence of the interplanetary magnetic field (IMF). The simulation is based on three-dimensional Lax-Wendroff explicit scheme, providing second-order accuracy in space and time. The ions produced from the nucleus of the comet will add considerable effects on the microstructure of the solar wind, thus severely altering its physical properties. The present simulation focuses on careful analysis of these properties by means of simulating the behavior of the comet Hale-Bopp’s tail at 1 AU from the sun. These properties include the changes of the plasma density, particle velocity, IMF changes, pressure, and internal energy. The results indicated that comet tail will highly disturb the physical properties of the solar wind for a considerable distance. These changes reflect the effect of including the source term in the present simulation. It is shown that the comet tail will retain its original shape faster as it approaches the sun. Practical comparisons are also presented in the present research with earlier work. The present simulation was made using MATLAB program.

Accuracy of Kinematic Wave and Diffusion Wave Approximations for Flood Routing. I: Steady Analysis

The applicability of the kinematic wave (KW) and diffusive wave (DW) approximations was investigated for steady flow in prismatic channels by using a second-order two-step Lax-Wendroff numerical scheme coupled with the characteristic method at the boundaries. Two types of downstream boundary conditions, critical-flow depth and zero-flow depth gradient, were considered together with the condition of discharge hydrograph reaching the steady state at the upstream end. The role of inertial, pressure, friction, and gravity forces was investigated for 16 test cases defined through the kinematic wave number K and the Froude number F0 , whose range was assumed to be (3–30) and (0.1–1), respectively. Errors were computed by comparing the steady dimensionless profiles of the flow depth with those estimated by the dynamic wave solution. The accuracy of the two approximations was assessed through the mean value e of the magnitudes of errors computed for the channel region where the solution was not significantly infl...

On the immersed interface method for solving time-domain Maxwell's equations in materials with curved dielectric interfaces

This paper deals with accurate numerical simulation of two-dimensional time-domain Maxwell's equations in materials with curved dielectric interfaces. The proposed fully second-order scheme is a hybridization between the immersed interface method (IIM), introduced to take into account curved geometries in structured schemes, and the Lax–Wendroff scheme, usually used to improve order of approximations in time for partial differential equations. In particular, the IIM proposed for two-dimensional acoustic wave equations with piecewise constant coefficients [C. Zhang, R.J. LeVeque, The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion 25 (1997) 237–263] is extended through a simple least squares procedure to such Maxwell's equations. Numerical results from the simulation of electromagnetic scattering of a plane incident wave by a dielectric circular cylinder appear to indicate that, compared to the original IIM for the acoustic wave equations, the augmented IIM with the proposed least squares fitting greatly improves the long-time stability of the time-domain solution. Semi-discrete finite difference schemes using the IIM for spatial discretization are also discussed and numerically tested in the paper.