Lax-Wendroff Method Research Articles

High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws

Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large class of time integrators include all popular (multistage) Runge–Kutta as well as single-step (multiderivative) Taylor methods. (The latter are commonly referred to as Lax–Wendroff methods when applied to PDEs). In this work, we offer explicit multistage multiderivative time integrators for hyperbolic conservation laws. Like Lax–Wendroff methods, multiderivative integrators permit the evaluation of higher derivatives of the unknown in order to decrease the memory footprint and communication overhead. Like traditional Runge–Kutta methods, multiderivative integrators admit the addition of extra stages, which introduce extra degrees of freedom that can be used to increase the order of accuracy or modify the region of absolute stability. We describe a general framework for how these methods can be applied to two separate spatial discretizations: the discontinuous Galerkin (DG) method and the finite difference essentially non-oscillatory (FD-WENO) method. The two proposed implementations are substantially different: for DG we leverage techniques that are closely related to generalized Riemann solvers; for FD-WENO we construct higher spatial derivatives with central differences. Among multiderivative time integrators, we argue that multistage two-derivative methods have the greatest potential for multidimensional applications, because they only require the flux function and its Jacobian, which is readily available. Numerical results indicate that multiderivative methods are indeed competitive with popular strong stability preserving time integrators.

Kinetics modeling of ampicillin nanoparticles synthesis via supercritical gas antisolvent process

Modeling of supercritical gas antisolvent (GAS) process was carried out for ampicillin nanoparticles synthesis. The particle size distribution of ampicillin limits bioavailability. Therefore, the kinetic data are essential for the control of particle size. Volume expansion and phase equilibrium modeling was studied to determine optimal operating conditions for experimental ampicillin production. Experimental ampicillin precipitations with GAS process at various antisolvent addition flow rates were investigated. The process model was then studied for the determination of nucleation and growth rate parameters. Equation of state, material and population balance equations were used for this modeling. A combination of the Crank-Nicholson and Lax-Wendroff methods was utilized to solve the population balance equation. Comparison of the experimental and modeling data showed that the model successfully predicted the particle size distribution. The effect of antisolvent addition rate on nucleation indicated that nucleation was enhanced via higher antisolvent addition rate and consequently smaller particle size was obtained. The mean particle sizes of ampicillin were obtained to be 357.09, 337.04 and 356.68nm at antisolvent flow rates of 1.6, 2 and 2.4mL/min, respectively.

An inverse Lax–Wendroff method for boundary conditions applied to Boltzmann type models

In this paper we present a new algorithm based on Cartesian meshes for the numerical approximation of kinetic models set in an arbitrary geometry. Due to the high dimensional property of kinetic models, numerical algorithms based on unstructured meshes are not really appropriate since most of numerical methods (semi-Lagrangian, spectral methods) are particularly efficient on structured grids. Here we propose to adapt the inverse Lax–Wendroff procedure, which has been recently introduced for conservation laws [21], for kinetic equations. Numerical simulations in 1D×3D and 2D×3D based on this approach are proposed for Boltzmann type operators (BGK, ES-BGK models).

Numerical simulation of water hammer in low pressurized pipe: comparison of SimHydraulics and Lax-Wendroff method with experiment

Article describes simulation of unsteady flow during water hammer with two programs, which use different numerical approaches to solve ordinary one dimensional differential equations describing the dynamics of hydraulic elements and pipes. First one is Matlab-Simulink-SimHydraulics, which is a commercial software developed to solve the dynamics of general hydraulic systems. It defines them with block elements. The other software is called HYDRA and it is based on the Lax-Wendrff numerical method, which serves as a tool to solve the momentum and continuity equations. This program was developed in Matlab by Brno University of Technology. Experimental measurements were performed on a simple test rig, which consists of an elastic pipe with strong damping connecting two reservoirs. Water hammer is induced with fast closing the valve. Physical properties of liquid and pipe elasticity parameters were considered in both simulations, which are in very good agreement and differences in comparison with experimental data are minimal.

An efficient reverse time migration method using local nearly analytic discrete operator

In recent years, reverse time migration (RTM), the most powerful depth imaging method, has become the preferred imaging tool in many geologic settings because of its ability to handle complex velocity models including steeply dipping interfaces. Finite difference is one of the most popular numerical methods applied in RTM in the industry. However, it often encounters a serious issue of numerical dispersion, which is typically suppressed by reducing the propagation grid sizes, resulting in large computation and memory increment. Recently, a nearly analytic discrete operator has been developed to approximate the partial differential operators, from which many antidispersion schemes have been proposed, and are confirmed to be superior to conventional algorithms in suppressing numerical dispersion. We apply an optimal nearly analytic discrete (ONAD) method to RTM to improve its accuracy and performance. Numerical results show that ONAD can be used effectively in seismic modeling and migration based on the full wave equations. This method produces little numerical dispersion and requires much less computation and memory compared to the traditional finite-difference methods such as Lax-Wendroff correction method. The reverse time migration results of the 2D Marmousi model and the Sigsbee2B data set show that ONAD can improve the computational efficiency and maintain image quality by using large extrapolation grids.

Dispersion and Dissipation Errors of Two Fully Discrete Discontinuous Galerkin Methods

The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K=kh?0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.

Mechanics of the foot Part 2: A coupled solid–fluid model to investigate blood transport in the pathologic foot

A coupled computational model of the foot consisting of a three-dimensional soft tissue continuum and a one-dimensional (1D) transient blood flow network is presented in this article. The primary aim of the model is to investigate the blood flow in major arteries of the pathologic foot where the soft tissue stiffening occurs. It has been reported in the literature that there could be up to about five-fold increase in the mechanical stiffness of the plantar soft tissues in pathologic (e.g. diabetic) feet compared with healthy ones. The increased stiffness results in higher tissue hydrostatic pressure within the plantar area of the foot when loaded. The hydrostatic pressure acts on the external surface of blood vessels and tend to reduce the flow cross-section area and hence the blood supply. The soft tissue continuum model of the foot was modelled as a tricubic Hermite finite element mesh representing all the muscles, skin and fat of the foot and treated as incompressible with transversely isotropic properties. The details of the mechanical model of soft tissue are presented in the companion paper, Part 1. The deformed state of the soft tissue continuum because of the applied ground reaction force at three foot positions (heel-strike, midstance and toe-off) was obtained by solving the Cauchy equations based on the theory of finite elasticity using the Galerkin finite element method. The geometry of the main arterial network in the foot was represented using a 1D Hermite cubic finite element mesh. The flow model consists of 1D Navier-Stokes equations and a nonlinear constitutive equation to describe vessel radius-transmural pressure relation. The latter was defined as the difference between the fluid and soft tissue hydrostatic pressure. Transient flow governing equations were numerically solved using the two-step Lax-Wendroff finite difference method. The geometry of both the soft tissue continuum and arterial network is anatomically-based and was developed using the data derived from visible human images and magnetic resonance images of a healthy male volunteer. Simulation results reveal that a two-fold increase in tissue stiffness leads to about 28% reduction in blood flow to the affected region.

Iterative Operator-splitting Methods with Embedded Discretization Schemes

In this paper we describe a computation of iterative operator-splitting method, which are known as competitive splitting methods, see [11] and [10]. We derived a closed form, based on commutators for the iterative method. The time discretization schemes apply extrapolation schemes and Pade approximations to the exp-functions. Spatial discretization schemes considered Lax-Wendroff methods and are combined with the iterative schemes. The error analysis describe the approximation errors. Numerical examples of ordinary and partial differential equations support the fast computation ideas.

Determination of time-varying pressure field from phase contrast MRI data

Methods We propose a numerical framework for determining pressure field in large arteries, the main components of which are: axi-symmetric 1-D unsteady wave-propagation model with elastic walls and an optimization framework for model personalization via parameter estimation from measured flow and anatomical data (Fig 1). 4D PC MRI data was acquired using a multi-slice phase contrast FLASH cine sequence with 3d velocity encoding (Siemens Magnetom Verio, temporal resolution 52 ms, matrix 100x192, FOV 319x219 mm, slice thickness 5 mm) in a sagittal to coronal slice orientation. Siemens 4D Flow WIP software was used for background phase, motion and venc anti-aliasing corrections, extracting vessel models including centerline trees and lumen boundary, and for automated extraction of flow profiles and cross-sectional areas. We use a linear elastic model for the vessel wall, with an axially varying stiffness. The parameters of the material model are estimated by a numerical optimization procedure, wherein the simulated deformation is fitted to the observed deformation of the vessel wall extracted from the MR data. Unsteady flow rate at the aortic inlet is extracted from PC-MRI data and used as the inlet boundary condition. To obtain physiologically valid mean and pulse pressure values and the wave propagation effects, we use a structured tree outflow boundary condition, wherein a morphological or fractal vessel tree is introduced to model the downstream behavior at the outlets of the vessel tree. The parameters for the outlet boundary condition are then independently perturbed at each individual outlet in order to match the measured unsteady outflow rate from PC-MRI data. This is achieved by adapting the final radius at which the asymmetric binary tree is terminated. The hyperbolic PDEs are solved using finite differences based on 2nd-order Lax-Wendroff method. Simulations converge after 3 cycles, with a maximum relative difference of 1% in pressure and flow between 2 consecutive cycles.

A Spectral Method in Time for Initial-Value Problems

A time-spectral method for solution of initial value partial differential equations is outlined. Multivariate Chebyshev series are used to represent all temporal, spatial and physical parameter domains in this generalized weighted residual method (GWRM). The approximate solutions obtained are thus analytical, finite order multivariate polynomials. The method avoids time step limitations. To determine the spectral coefficients, a system of algebraic equations is solved iteratively. A root solver, with excellent global convergence properties, has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by use of temporal and spatial subdomains. As examples of advanced application, stability problems within ideal and resistive magnetohydrodynamics (MHD) are solved. To introduce the method, solutions to a stiff ordinary differential equation are demonstrated and discussed. Subsequently, the GWRM is applied to the Burger and forced wave equations. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Thus the method shows potential for advanced initial value problems in fluid mechanics and MHD.

A flux-limiter method for dam-break flows over erodible sediment beds

Finite volume methods for dam-break flows over erodible sediment beds require a monotone numerical flux. In the present study we present a new flux-limiter scheme based on the Lax–Wendroff method coupled with a non-homogeneous Riemann solver and a flux limiter function. The non-homogeneous Riemann solver consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The proposed method satisfy the conservation property such that the discretization of the flux gradients and the source terms are well-balanced in the numerical solution of suspended sediment models. The flux-limiter method provides accurate results avoiding numerical oscillations and numerical dissipation in the approximated solutions. Several standard test examples are considered to verify the performance and the accuracy of the proposed method.

Analysis of shock capturing methods for chemical species transport in unsteady compressible flow

This paper presents a chemical species transport model to account for variable composition and gas properties along the flow path in internal combustion engines. The numerical solution to adapt the gas dynamic model to chemical species transport in boundary conditions by means of the Method of Characteristics and in volumes by means of a filling and emptying model is described. The performance for chemical species transport in 1D elements of shock-capturing methods, such as the two-step Lax–Wendroff method and the Sweby’s TVD scheme considering several flux limiter definitions, is carried out by means of shock-tube tests. The influence of the fluid modelling as perfect or non-perfect gas on the numerical methods features and the flow characteristics on shock-tube results are analysed.

A stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space for the scalar wave equation

Based on the formula for stability of finite-difference methods with second-order in time and general-order in space for the scalar wave equation, I obtain a stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space. Unlike the formula for methods with second-order in time, this formula depends on two parameters: one parameter is related to the weights for approximations of second spatial derivatives; the other parameter is related to the weights for approximations of fourth spatial derivatives. When discretizing the mixed derivatives properly, the formula can be generalized to the case where the spacings in different directions are different. This formula can be useful in high-accuracy seismic modeling using the scalar wave equation on rectangular grids, which involves both high-order spatial discretizations and high-order temporal approximations. I also prove the instability of methods obtained by applying high-order finite-difference approximations directly to the second temporal derivative, and this result solves the “Bording’s conjecture.”

A Non-dimensional Form of Hydrodynamic Model with Variable coefficients in a Uniform Reservoir Using Lax-Wendroff Method

A mathematical model is used to simulate the water current and the elevation in a uniform reservoir. A hydrodynamic model that provides the velocity field and elevation of the water flow is considered. In the simulating process, the Lax-Wendroff technique is used to approximate the solutions.

A Split-Step Algorithm for Effectively Suppressing the Numerical Dispersion for 3D Seismic Propagation Modeling

In this article, we propose a 3D split-step algorithm (SSA) for seismic- wave simulation. We first transform the wave equations in 3D anisotropic media into a system of first-order partial differential equations with respect to time t. Then we use the multidimensional high-order interpolation method to approximate the high-order spatial derivatives, so that we obtain a system of semidiscrete ordinary differential equations (ODEs). Finally, the third-order implicit Adams method and truncated dif- ferentiator series method are applied to solve the semidiscrete ODEs. We provide the theoretical study on the properties of the 3D SSA, such as stability criteria, theoretical error, numerical error, numerical dispersion, and computational efficiency. We also compare some seismic modeling results of this method against those of some high-order finite-difference schemes. Theoretical analysis and numerical tests show that the 3D SSA is third-order accurate in time and fourth-order accurate in space. However, its computational costs and memory requirements are much less than those of the fourth-order Lax-Wendroff correction method and the fourth-order staggered- grid method. Using a multilayer elastic model with large velocity contrasts and free surface, we compare the result of the 3D SSA with that of the discrete-wavenumber method. We also present the synthetic seismograms in the 3D three-layer isotropic medium, the wave-field snapshots in the 3D two-layer medium, and the 3D trans- versely isotropic medium with a vertical symmetry axis. All these promising numer- ical results illustrate that the 3D SSA can suppress effectively the numerical dispersion caused by discretizing the wave equations when too few sampling points per mini- mum wavelength are used or when models have large velocity contrasts between adjacent layers, further resulting in both increasing the computational efficiency and saving the storage space when big grids are used.

Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping

SUMMARY We investigate the stability of some high-order finite element methods, namely the spectral element method and the interior-penalty discontinuous Galerkin method (IP-DGM), for acoustic or elastic wave propagation that have become increasingly popular in the recent past. We consider the Lax-Wendroff method (LWM) for time stepping and show that it allows for a larger time step than the classical leap-frog finite difference method, with higher-order accuracy. In particular the fourth-order LWM allows for a time step 73 per cent larger than that of the leap-frog method; the computational cost is approximately double per time step, but the larger time step partially compensates for this additional cost. Necessary, but not sufficient, stability conditions are given for the mentioned methods for orders up to 10 in space and time. The stability conditions for IP-DGM are approximately 20 and 60 per cent more restrictive than those for SEM in the acoustic and elastic cases, respectively.

A water quality computation in the uniform reservoir

In this research, two mathematical models are used to simulate pollution due to sewage effluent in the uniform reservoir with varied current velocity. The first is the hydrodynamic model that provides the velocity field and elevation of the water flow. The second is the dispersion model that gives the pollutant concentration fields. In the simulating processes, we used the Lax-Wendroff method to system of hydrodynamic model and the forward in time and backward in space technique to the dispersion model. The accuracy of the models is tested by the illustrative example.

A discretized population balance for particle formation from gas antisolvent process: The combined Lax-Wendroff and Crank-Nicholson method

Population balance models describing particle formation kinetics in supercritical crystallization processes in general and of gas antisolvent crystallization (GAS) process in particular are rather complicated. Depending on the considered particle formation mechanism, population balance model of GAS process may include phenomena such as primary nucleation, secondary nucleation, crystal growth, as well as agglomeration and/or breakage (attrition) of crystals. The resulting equation is often of the integro-partial differential form. A powerful numerical algorithm for the treatment of the implemented population balance model structures is presented in this paper. The method/ algorithm is a combination of the Lax-Wendroff and Crank-Nicholson methods. Algorithm simulations were performed for changes in the main GAS process operating parameters, i.e., the antisolvent addition rate and saturation level. The simulations were performed at a process temperature of 25 °C, while the antisolvent addition rate, Q A , was varied between 1 and 100 ml/min, and the initial solute concentration was varied between 25% and 100% of the concentration ratio. The numerical algorithm was successfully able to treat the implemented population balance model structures, producing smooth dynamic and steady state particle size distributions. Moreover, simulation findings were reasonably consistent with the experimentally obtained data.

Basic design scheme for wave rotors

Pressure wave devices use shock waves to transfer energy directly between fluids without additional mechanical components, thus having the potential for increased efficiency. The wave rotor is a promising technology which uses shock waves in a self-cooled dynamic pressure exchange between fluids. For high-pressure, high-temperature topping cycles, it results in increased engine overall pressure and temperature ratio, which in turn generates higher efficiency and lower specific fuel consumption. Designing a wave rotor mainly focuses on predicting the behavior of shock and expansion waves. The extant literature presents numerous examples of wave rotor designs, but most of them rely on complicated numerical analyses as well as computer code developed specifically for this application. This paper presents an initial scheme used for designing wave rotors employing thermodynamic and gasdynamic analysis as well as computational fluid dynamic analysis. Basic theory and a simplified model of the wave rotor are used to predict the travel time and strength of waves. The model is then refined using a more advanced numerical scheme on the basis of the Lax–Wendroff method and FLUENT, a commercial CFD code.

Time dependent monochromatic scattering of radiation in one dimensional media: Analytical and numerical solutions

In order to choose a numerical method for solving the time dependent equations of radiative transport, we obtain an exact solution for the time dependent radiation field in a one dimensional infinite medium with monochromatic, isotropic scattering for sources with an arbitrary spatial distribution and an arbitrary time variation of their power. The Lax-Wendroff method seems to be the most suitable. Because it is assumed that radiation delay is caused by the finite speed of light, the following difficulty arises when the numerical method is used: the region of variation of the variables (position τ and time t) is triangular (the inequality τ ≤ t). This difficulty is overcome by expanding the unknown functions in series in terms of small values of the time and position. By comparing the numerical and exact solutions for a point source with a given time dependence for its power and with pure scattering, the number of steps in the variables required to obtain a desired accuracy is estimated. This numerical method can be used to calculate the intensity and polarization of the radiation from sources in the early universe during epochs close to the recombination epoch.