Lax-Wendroff Scheme Research Articles

Lax-Wendroff Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected To Integral Condition

In this paper, the Lax-Wendroff difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion and the Lax-Wendroff difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error,    and    norm.  The numerical results presented in tables and

A fast method for solving quasi-pentadiagonal Toeplitz linear systems and its application to the Lax–Wendroff scheme

In this paper we present an algorithm for solving quasi-pentadiagonal Toeplitz linear systems, that is based on Du et al. method (Du et al., 2014) and the Sherman–Morrison–Woodbury inversion formula. All algorithms have been implemented in Matlab and numerical experiments are given in order to illustrate the validity and efficiency of our algorithm. An application of the proposed algorithm to the Lax–Wendroff scheme is also considered.

Analysis of Aeroacoustic Properties of the Local Radial Point Interpolation Cumulant Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has recently been used to simulate wave propagation, one of the challenging aspects of wind turbine modeling and simulation. However, standard LB methods suffer from the instability that occurs at low viscosities and from its characteristic lattice uniformity, which results in issues of accuracy and computational efficiency following mesh refinement. The local radial point interpolation cumulant lattice Boltzmann method (LRPIC-LBM) is proposed in this paper to overcome these shortcomings. The LB equation is divided into collision and streaming steps. The collision step is modeled by the cumulant method, one of the stable LB methods at low viscosities. In addition, the streaming step, which is naturally a pure advection equation, is discretized in time and space using the Lax–Wendroff scheme and the local radial point interpolation method (RPIM), a mesh free method. We describe the propagation of planar acoustic waves, including the temporal decay of a standing plane wave and the spatial decay of a planar acoustic pulse. The analysis of these specific benchmark problems has yielded qualitative and quantitative data on acoustic dispersion and dissipation, and their deviation from analytical results demonstrates the accuracy of the method. We found that the LRPIC-LBM replicates the analytical results for different viscosities, and the errors of the fundamental acoustic properties are negligible, even for quite low resolutions. Thus, this method may constitute a useful platform for effectively predicting complex engineering problems such as wind turbine simulations, without parameter dependencies such as the number of points per wavelength Nppw and resolution σ or the detrimental effect caused by the use of coarse grids found in other accurate and stable LB models.

Monotonicity-Preserving Lax–Wendroff Scheme for Solving Scalar Hyperbolic Conservation Laws

In this paper, we construct the monotonicity-preserving Lax–Wendroff (MP-Lax–Wendroff) scheme based on the MP scheme as proposed by Suresh and Huynh [11], which is a high-order and high-resolution method for hyperbolic conservation laws. It is well known that the total-variation-diminishing (TVD) methods possess the order reduction at smooth extremum points. However, the MP scheme not only preserves the non-oscillatory behavior of the procedure, but also prevents the order reduction of the method. We provide an analysis of the MP procedure using the numerical Lax–Wendroff flux sophistically in detail. Due to lack of robustness of the MP scheme, a rigorous and efficient MP-Lax–Wendroff scheme is introduced to enhance the robustness of the MP process. This means that a TVD numerical flux is applied to damp the oscillations at disturbed discontinuity points to get the high-resolution property. Therefore, we apply this idea to construct the improved MP-Lax–Wendroff scheme (we call it MP-R-Lax–Wendroff scheme). The computational performance of the MP-R-Lax–Wendroff scheme for linear and nonlinear hyperbolic conservation laws is conducted through some prototype examples which in turn validate our theoretical results.

CELL-CENTERED LAGRANGIAN LAX-WENDROFF HLL HYBRID SCHEME ON UNSTRUCTURED MESHES

We have recently introduced a new cell-centered Lax-Wendroff HLL hybrid scheme for Lagrangian hydrodynamics [Fridrich et al. J. Comp. Phys. 326 (2016) 878-892] with results presented only on logical rectangular quadrilateral meshes. In this study we present an improved version on unstructured meshes, including uniform triangular and hexagonal meshes and non-uniform triangular and polygonal meshes. The performance of the scheme is verified on Noh and Sedov problems and its second-order convergence is verified on a smooth expansion test.Finally the choice of the scalar parameter controlling the amount of added artificial dissipation is studied.

Lax-Wendroff methods on highly non-uniform meshes. Dedicated to the Memory of Blair Swartz (1932–2019)

The standard Lax-Wendroff scheme with the conservative Lax-Friedrichs nodal predictor on highly non-uniform meshes produces serious oscillations, making it useless on such meshes. Wendroff and White (1989) [13] proposed two versions (WW and WWJp) with different predictors which work robustly on such meshes. Both WW and WWJp are second order accurate. We investigate how these methods behave on highly non-uniform meshes of three types (Pike, cluster and van der Corput) for 1D smooth solutions of the Burgers and the Euler equations. The WW and WWJp methods are extended to 2D and tested on smooth solutions of the Euler equations on 2D meshes created by the Cartesian product of 1D highly non-uniform meshes. We have not been able to find any significant difference between the WW and WWJp results, thus the simpler WW should be preferred.

A Simple Solver for the Two-Fluid Plasma Model Based on PseudoSpectral Time-Domain Algorithm

We present a solver of 3D two-fluid plasma model for the simulation of short-pulse laser interactions with plasma. This solver resolves the equations of the two-fluid plasma model with ideal gas closure. We also include the Bhatnagar-Gross-Krook collision model. Our solver is based on PseudoSpectral Time-Domain (PSTD) method to solve Maxwell's curl equations. We use a Strang splitting to integrate Euler equations with source term: while Euler equations are solved with a composite scheme mixing Lax-Friedrichs and Lax-Wendroff schemes, the source term is integrated with a fourth-order Runge-Kutta scheme. This two-fluid plasma model solver is simple to implement because it only relies on finite difference schemes and Fast Fourier Transforms. It does not require spatially staggered grids. The solver was tested against several well-known problems of plasma physics. Numerical simulations gave results in excellent agreement with analytical solutions or with previous results from the literature.

High order numerical schemes for transport equations on bounded domains

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

Macroscopic Modelling of Pedestrian Flows Based on Conservation Law

Overcrowded of sidewalks can reduce the level of satisfaction of pedestrians. The large crowd on the sidewalks results in slowing down time of the pedestrian to arrive in their destination. This studt, focuses on pedestrian flows modelling using the macroscopic model. Numerical approximation of the macroscopic model is formulated as scalar hyperbolic conservation laws. The Lax-Wendroff scheme is used to discretize the equation of conservation laws. The simulation results show that the numerical approximation in term of density confirms the exact solution. In this simulation, the velocity function is obtained by curve fitting of observation data using linear regression method. The observation data, which are consist of velocity-density relation, are obtained from observation of pedestrian flows. The study case of this research conducts on the sidewalks of Braga Street, Bandung, Indonesia. There are two velocity functions used in the simulation, i.e. c 1 = 0.58206 + −0.94476ρ and c 2 = 0.59926 + −0.78052ρ, respectively. In performing the velocity function c 2, the pedestrian leader position is approximately 1 meter in front of the pedestrian leader using the velocity function c 1 at final time T = 20 seconds and T = 30 seconds. Overall, the numerical experiment shows that the pedestrian leader using the velocity functions c 2 is faster than by using c 1.

A Novel multidimensional Boltzmann neutrino transport scheme for core-collapse supernovae.

We introduce a new discrete-ordinate scheme for solving the general relativistic (GR) Boltzmann transport equation in the context of core-collapse supernovae (CCSNe). Our algorithm avoids the need to spell out the complicated advection terms in energy and angle that arise when the transport equation is formulated in spherical polar coordinates, in the comoving frame, or in a GR space-time. We instead approach the problem by calculating the advection of neutrinos across momentum space using an intuitive particle-like approach that has excellent conservation properties and fully accounts for Lorentz boosts, GR effects, and grid geometry terms. In order to avoid the need for a global implicit solution, time integration is performed using a locally implicit Lax-Wendroff scheme that correctly reproduces the diffusion limit. This will facilitate the use of our method on massively parallel distributed-memory architectures. We have verified the accuracy and stability of our scheme with a suite of test problems in spherical symmetry and axisymmetry. To demonstrate that the new algorithm works stably in CCSN simulations, we have coupled it to the GR hydrodynamics code coconut and present a first demonstration run of a [Formula: see text] progenitor with a reduced set of neutrino opacities.

Cell-centered Lagrangian Lax-Wendroff HLL hybrid scheme in cylindrical geometry

Lagrangian hydrodynamics as described by the Euler equations is treated by an improved version of the basic predictor-corrector Lax-Wendroff method that also has added HLL-type dissipative fluxes, including both artificial viscosity and energy dissipation. The method in Cartesian geometry is enhanced by a different weighting of the conservative variables in the predictor and a new treatment of material interfaces. The basic method is extended to cylindrical r,z geometry, where it satisfies the geometric conservation law and keeps exact spherical symmetry on equiangular polar meshes. The added viscosity does not preserve symmetry, so in order to achieve that we have added a symmetry correction. Numerical hydrodynamic tests, including the Noh, Sedov, spherical Sod, free expansion and Kidder problems show a reasonable performance of the method. In addition, the spherical Noh problem was simulated on an initially rectangular mesh with a very good result regarding its symmetry.

Modeling and assessment of two-phase transonic steam flow with condensation through the convergent-divergent nozzle

In the Fukushima accident, it was found that the Reactor Core Isolation Cooling (RCIC) system played a crucial role in delaying core meltdown by almost three days in Fukushima Daiichi Unit 2, because of self-regulated operation of the steam driven RCIC turbine-pump injection system. Steam flow in the convergent-divergent nozzles of the RCIC Terry turbine is two-phase non-equilibrium transonic flow with homogenous nucleation condensation. To more accurately predict the dynamic process and behavior of the transonic compressible steam flow, a one-dimensional transient two-phase analytical model is presented. A simplified four-fluid model was employed in the present work with the consideration of four separate fluid fields: vapor, liquid film, entrained droplets and condensed droplets. The mass, momentum and energy interactions between the fluids were considered and modeled. An extended seven-equation non-equilibrium critical flow model was developed to obtain the critical pressure and velocities of each phase at the nozzle throat. To predict the wetness in the divergent section, a mechanistic nucleation condensation model was integrated in the nozzle analysis model, considering the generation and consequent growth of droplets. The governing differential equations on a staggered grid were discretized using the second-order Lax-Wendroff scheme with a flux limiter, and the Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was employed to solve the discrete linear system. To demonstrate the predictability and reliability of the physical models and the numerical method proposed in the present work, three representative nozzles were modeled and simulated. The results show good agreement with the available experimental data, even for condensation shock.

Calculation of Interphase Mass Transfer in a Spray Flow Produced by a Nozzle with Account of Crisis

This paper describes the numerical model and the results of calculating the interphase mass transfer in a two-phase flow generated by atomizing a liquid in a gas using a nozzle. The proposed mathematical model is based on differential equations of unsteady flow in a compressible medium, complemented by the equation of mass transfer from gas to droplets. The popular explicit Lax–Wendroff scheme is used for transition to difference analogs of continuity and phase motion equations. Axial profiles of droplet and gas velocities and gas impurity concentrations in a free spray cone are calculated and presented together with radial profiles of impurity concentrations in a two-phase flow through a cylindrical apparatus, taking into account early drag crisis of droplets and crisis of interphase mass transfer, as well as the features of turbulent friction in gas, discovered in previous experiments. The calculation established the dependencies of the gas flow rate, the concentration of gas impurities at the outlet of the apparatus, and the amount of impurities absorbed by the liquid from the height and cross-sectional area of the apparatus.

Lax-Wendroff schemes for elastic-plastic solids

Two formulations of the Lax-Wendroff scheme are proposed in this paper for its extension to elastic-plastic solids. The unknown vector consists of the strain or stress components in addition to the velocity ones according to the chosen formulation. The Lax-Wendroff scheme is here implemented in its Richtmyer two-step version so that the projection of the elastic trial stresses onto the yield locus is performed twice per time step. It is shown that it allows to reduce the number of cells required for the approximation of discontinuous plastic waves with respect to a classical one-step strategy consisting in an a posteriori projection of the elastic trial stresses onto the yield locus. Elastic-plastic associated and non-associated constitutive models under small strains are considered in examples. In particular, dynamic ratchetting is simulated with the Armstrong-Frederick nonlinear kinematic hardening. Comparisons with respect to finite element numerical solutions show good agreements.

Conditions for L2-Dissipativity of Linearized Explicit Difference Schemes with Regularization for 1D Barotropic Gas Dynamics Equations

Explicit two-level in time and symmetric in space difference schemes constructed by approximating the 1D barotropic quasi-gas-/quasi-hydrodynamic systems of equations are studied. The schemes are linearized about a constant solution with a nonzero velocity, and, for them, necessary and sufficient conditions for the $${{L}^{2}}$$ -dissipativity of solutions to the Cauchy problem are derived depending on the Mach number. These conditions differ from one another by at most twice. The results substantially develop the ones known for the linearized Lax–Wendroff scheme. Numerical experiments are performed to analyze the applicability of the found conditions in the nonlinear formulation to several schemes for different Mach numbers.

Development of a three dimensional meshless numerical method for the solution of the Boltzmann equation on complex geometries

We propose a three dimensional meshless numerical scheme based on the Boltzmann equation for the simulation of fluid flows in domains of complex geometries. The velocity space discretization in our method is the same as the standard lattice Boltzmann method, i.e. the same lattices are used to discretize microscopic velocities. The velocity-discrete Boltzmann equation with the BGK collision approximation is discretized in time using the Lax–Wendroff scheme, and subsequently in space using the meshless local Petrov–Galerkin method based on local radial basis functions augmented with polynomials. The method is first validated and tuned by solving two benchmark problems: The pressure-driven flow in an elliptical tube, and the three-dimensional lid-driven cavity flow. The error analysis for the first case shows the second order accuracy of the proposed method. The flow in a packed bed is then solved in order to illustrate the efficiency of the proposed method in the simulation of geometrically complex flows. The results show that both the computational time and the memory usage of our method are lower than those of the standard lattice Boltzmann method for the geometrically complex flows.

Numerical study of steady-state acoustic oscillations in semi-closed straight channel

The influence of properties of finite-difference schemes of Godunov, Lax-Wendroff and TVD on prediction of amplitude-frequency responses of pressure oscillations is investigated as a part of a problem of steady-state oscillations in a semi-closed channel. Problems of Riemann and pipe draining are solved as a test. It is shown that the dissipative properties of Godunov scheme underestimate the amplitude of high-frequency oscillations relative to the experimental data. Lax-Wendroff scheme predicts an amplitude-frequency response with sufficient degree of accuracy. TVD scheme leads to nonmonotonic amplitude-frequency response and overestimates values of resonance frequencies.

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

Improvement of one-dimensional gas dynamic model for pulsation prediction in reciprocating compressor systems

Employing gas dynamic model for simulation of the transient fluid flow is critical for compression system in the design process to avoid critical piping layout, and also in operation for troubleshooting. The two-step Lax-wendroff (LW2) scheme has been usually utilized for speed and stability. This paper extends the original LW2 scheme to a new one with the improvement of bringing less dispersion and computational time. An iterative methodology is designed for the Trapezoidal integration of the Riemann variables at pipe end to improve the boundary condition calculation. The acoustic behavior of a simple geometry is computed by the two LW2 numerical schemes. Comparing the predicting results with the published experimental data validates the refinement of this new LW2 scheme. The instantaneous pulsating pressures obtained from numerical calculations are also compared with experimental measurements in the discharge system of a reciprocating compressor, with fair agreement.

On Conditions for L2-Dissipativity of Linearized Explicit QGD Finite-Difference Schemes for One-Dimensional Gas Dynamics Equations

An explicit two-level in time and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L2-dissipativity of solutions to the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity and depending on the Mach number. It is shown that the condition on the Courant number can be made independent of the Mach number. The results provide a substantial development of the well-known stability analysis of the linearized Lax–Wendroff scheme.