Lax-Wendroff Scheme Research Articles

A lightweight, conservation-moment-based implicit gas kinetic Lax–Wendroff scheme for all Knudsen number isothermal gas flows

This work presents an efficient implicit gas kinetic Lax–Wendroff scheme for steady isothermal gas flows in all Knudsen number (Kn) regimes. In the scheme, the discrete velocity Bhatnagar–Gross–Krook model equations (DVE) and the associated conservation moment equations (CME) are coupled and solved by matrix-free implicit schemes. Thanks to obtaining the fluxes of the CME by multiplying the fluxes of the DVE with a projection matrix and utilizing the equilibrium distribution functions at the new time predicted by the CME, both the complicated macro fluxes reconstruction of the CME and the calculation of the Jacobian matrix of the equilibrium distribution functions are not needed in the scheme, which makes the scheme lightweight. Moreover, to enhance the accuracy of the predicted equilibrium distribution functions at the new time to improve the convergence, a symmetric Gauss–Seidel scheme with inner iterations is used to solve the CME system. Due to the coupling between the DVE and the CME, the highly efficient implicit scheme for the CME drives the DVE system to converge quickly for the continuum and near-continuum flows. Furthermore, to verify the accuracy and high efficiency of the proposed implicit scheme, comparison studies of several two-dimensional isothermal rarefied gas flow cases simulated by the present implicit scheme and the explicit gas kinetic Lax–Wendroff scheme are also provided. The numerical results show that the present implicit scheme can be as accurate as its explicit counterpart with one to two orders times speed-up in all Kn number flow regimes.

A new compact scheme-based Lax–Wendroff method for high fidelity simulations

The classical Lax–Wendroff (LW) method employs explicit second order central difference schemes to solve partial differential equations. Recently, the authors in Sengupta et al. (2023) have identified numerical parameter ranges in performing large eddy simulation (LES) using this classical LW method. In a bid to improve these numerical parameter ranges for high fidelity computations, a new non-uniform grid based compact scheme for LW method (NUCLW) is developed here. This new method uses sixth order non-uniform compact scheme (NUC6) for the first and higher order derivatives. Reported global spectral analysis confirms substantial improvement in the resolution and accuracy of NUCLW scheme over the classical LW scheme. A significant advantage of the present scheme is its ability to compute solutions for non-uniform, structured grids, and that does not suffer degradation in accuracy. The potential of the NUCLW scheme for high accuracy computations is demonstrated by solving 2D incompressible Navier–Stokes equations for the benchmark unsteady flow inside a square lid driven cavity problem for a subcritical Reynolds number of 3200 and a post-critical Reynolds number of 10,000 and simulating the instability of the Taylor–Green vortex problem. These demonstrate the ability of the NUCLW scheme to solve the Navier–Stokes equations for high fidelity simulations such as LES/DNS with enhanced ranges of numerical parameters with respect to enhanced accuracy and faster computations.

Development of transient thermal-hydraulic analysis model for the two-phase loop thermosyphon

To comprehensively and accurately simulate and predict the flow and heat transfer characteristics of the two-phase loop thermosyphon (TPLT), a two-fluid two-phase model was developed. A six-equation set was formulated to account for two-phase flow, encompassing mass equations, momentum equations, and energy equations. The interactions between the fluids in terms of mass, momentum, and energy were also taken into account. To obtain numerical solutions for the partial differential equations within a staggered mesh framework, we employed the 2nd-order Lax-Wendroff scheme to obtain the discrete linear system, complemented by the inclusion of a flux limiter. The Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was employed to address the solution of the linear equation system. Data from a small-scale TPLT test was used to assess the accuracy and reliability of both the physical models and the numerical method proposed in this study. Simulations were performed under two working conditions of the thermosyphon with filling ratio of 45 % and 64 %. The simulation results exhibited good agreement with the experimental data, demonstrating the accuracy of the model. Furthermore, the model is capable of providing information on the complex two-phase flow inside the TPLT that cannot be measured experimentally.

A New Algorithm For Solving Thermal Newtonian Flow In Axisymmetric Straight Channel

In this study, three main problems in fluid flow have been discussed. First, the derivation of the energy equation is conducted within cylindrical coordinates, which has not been addressed by the researchers before. Secondly, we develop a new algorithm for solving the governing equations that describe incompressible Newtonian flows under thermal condition. This algorithm is constructed based on the so-called Taylor-Galerkin/Pressure-Correction finite element method with the help of the two-step Lax-Wendroff scheme. Finally, we study the effect of non-dimensional numbers, such as the Reynolds number (Re) and the Prandtl number (Pr), on the velocity, pressure, and temperature. The results show that the developed method is consistent with established physical phenomena and other studies

Cell-centered Lagrangian Lax-Wendroff HLL hybrid scheme in 3D

This paper describes a cell-centered Lax-Wendroff type scheme with artificial dissipation for Lagrangian fluid dynamics on general unstructured 3D meshes. The method is an extension of a similar 2D approach to 3D. The dissipative terms are inspired by the finite volume HLL flux formulation and are used as artificial viscosity in the momentum equation and as artificial energy flux in the energy equation. Special treatment is formulated for contact discontinuities and mesh-aligned interfaces. The proposed second order method provides reasonable results for typical 3D hydrodynamic tests, such as Noh, Sedov, Kidder and spherical Sod problems.

Numerical study of Newtonian laminar Flow around circular and square cylinders

In this study, we present a novel algorithm designed to address the computational challenges posed by the governing equations of incompressible Newtonian fluid flow, particularly of bluff bodies of circular and square in a channel. This a significant innovation algorithm is developed based on the Taylor–Galerkin/Pressure-Correction finite element method. Notably, a pivotal element of this approach involves the implementation of the two-step Lax–Wendroff scheme to resolve the momentum equation. To assess the efficacy of our novel algorithm, we conducted a comprehensive analysis using a standard benchmark scenario; the initiation of Poiseuille flow through an axisymmetric rectangular channel encompassing bluff bodies. In this context, our investigation centered on the influence of the Reynolds number (Re) on various pertinent variables. The results obtained from our study underscore the remarkable congruence between the outcomes generated by the new algorithm and established physical principles, as well as findings from prior research, as evidenced through thorough comparisons with the existing body of literature. This research represents a significant stride toward advancing our understanding of fluid dynamics and computational methodologies in this domain.

Propagation characteristics of tsunami waves in shear flow based on the Boussinesq model with constant vorticity

Propagation of tsunami wave in linear shear flow with constant vorticity is studied in this paper. The nonlinear and dispersive Boussinesq equations for water waves in shear flow are derived by introducing uniform vorticity into the Euler equations. Generally, the transoceanic propagation of tsunami can be well simulated by the weakly nonlinear Boussinesq model. Thus, the nonlinear and dispersive terms are retained to the first-order and second-order (ϵ,μ2) respectively in the Boussinesq equations with uniform vorticity in the present work, which satisfies the simulation accuracy especially the dispersion effects of the long-distance propagation of tsunamis and does not require huge computational cost for the simulation. The set of equations is solved with the total variation diminishing (TVD) Lax–Wendroff scheme which is second-order both in space and time. This TVD scheme is applicable to handle the dispersion effects of the N-waves and undular bores. Different patterns of tsunami wave in the water with presence of vorticity are simulated by the numerical model. Wave height and wavelength are affected by the current with vorticity. The pattern of solitary wave is significantly changed in upstream and downstream conditions. The wave deformation and propagation speed of N-waves and undular bores in the shear flow are also studied. The vorticity has impacts on the leading crest amplitude and the maximum wave appearance time for N-wave and changes the wave amplitude and the following water level for undular bores.

The Von Neumann conditions for the upwind and Lax-Wendroff schemes of the Telegraph Equation

In this paper, the Von Neumann conditions for the Upwind and Lax-Wendroff schemes of the Telegraph Equation are derived, but then prove to be contradicted with the proportional relation between time and space steps, determined by voltage and current waves’ propagation velocity. The contradiction implies that the stable Upwind and Lax-Wendroff schemes of the Telegraph Equation are not physically applicable.

A phase-field Lattice Boltzmann method for liquid-vapor phase change problems based on conservative Allen-Cahn equation and adaptive treegrid

A phase-field Lattice Boltzmann method (LBM) based on Allen-Cahn (A-C) equation with a modified adaptive mesh refinement (AMR) method is presented to simulate the liquid-vapor phase change problem. Three Lattice Boltzmann equations are solved: one for the conservative Allen-Cahn equation, one for the incompressible Navier-Stokes equations, and one for the energy equation. These three equations are written in the form of multiple-relaxation-time, together with the equilibrium and forcing terms using the set of Hermite polynomials. The adaptive mesh refinement technique with quadtree grid data structure is implemented to capture the liquid-vapor interface in simulations, in which the mesh can be locally refined by subdividing blocks identified by user-defined criterion. The Lax-Wendroff finite-difference scheme is adopted to calculate the whole computational domain continuously under nonuniform grids. The present method is validated by the classical Stefan problem and then performed to simulate the film boiling heat transfer. The numerical experiments verify the ability of the proposed method to capture the correct evolution of the interface, guarantee the numerical accuracy and remarkably save the computational cost.

Verification of a High-Order FEM-based CFD Code using the Method of Manufactured Solutions

A high-order computational fluid dynamics (CFD) code capable of solving compressible turbulent flow problems was developed. The CFD code employs the Flowfield Dependent Variation (FDV) scheme implemented in a Finite Element Method (FEM) framework. The FDV scheme is basically derived from the Lax-Wendroff Scheme (LWS) involving the replacement of LWS’s explicit time derivatives with a weighted combination of explicit and implicit time derivatives. The code utilizes linear, quadratic and cubic isoparametric quadrilateral and hexahedral Lagrange finite elements with corresponding piecewise shape functions that have formal spatial accuracy of second-order, third-order and fourth-order, respectively. In this paper, the results of observed order-of-accuracy of the implemented FDV FEM-based CFD code involving grid and polynomial order refinements on uniform Cartesian grids are reported. The Method of Manufactured Solutions (MMS) is applied to governing 2-D Euler and Navier-Stokes equations for flow cases spanning both subsonic and supersonic flow regimes. Global discretization error analyses using discrete 𝐿2 norm show that the spatial order-of-accuracy of the FDV FEM-based CFD code converges to the shape function polynomial order plus one, in excellent agreement with theory. Uniquely, this procedure establishes the wider applicability of MMS in verifying the spatial accuracy of a high-order CFD code.

The Neumann boundary condition for the two-dimensional Lax–Wendroff scheme

The Neumann boundary condition for the two-dimensional Lax–Wendroff scheme

Numerical Modeling of Pollutant Transport: Results and Optimal Parameters

In this work, we used three finite difference schemes to solve 1D and 2D convective diffusion equations. The three methods are the Kowalic–Murty scheme, Lax–Wendroff scheme, and nonstandard finite difference (NSFD) scheme. We considered a total of four numerical experiments; in all of these cases, the initial conditions consisted of symmetrical profiles. We looked at cases when the advection velocity was much greater than the diffusion of the coefficient and cases when the coefficient of diffusion was much greater than the advection velocity. The dispersion analysis of the three methods was studied for one of the cases and the optimal value of the time step size k, minimizing the dispersion error at a given value of the spatial step size. From our findings, we conclude that Lax–Wendroff is the most efficient scheme for all four cases. We also show that the optimal value of k computed by minimizing the dispersion error at a given value of a spacial step size gave the lowest l2 and l∞ errors.

A gas kinetic Lax–Wendroff scheme for low-speed isothermal rarefied gas flows

Previously, a gas kinetic Bhatnagar–Gross–Krook (BGK) scheme was proposed by us for incompressible flows in the continuum limits. [W. Li and W. Li, “A gas-kinetic BGK scheme for the finite volume lattice Boltzmann method for nearly incompressible flows,” Comput. Fluids 162, 126–138 (2018).] In the present work, we extend the gas kinetic BGK scheme to simulate low-speed isothermal rarefied nonequilibrium gas flows. This scheme is a gas kinetic Lax–Wendroff scheme (GKLWS) for the discrete velocity Boltzmann equation in the finite volume discretization framework with second-order accuracy in both time and space. As collision and transport of the molecular particles are coupled in the present GKLWS, the time step of the present method is not limited by the relaxation time, for which the present scheme is efficient for multiscale gas flows. Moreover, the present GKLWS holds the asymptotic preserving (AP) property, which ensures that both the Navier–Stokes solutions in the continuum limits and free-molecular flow solutions in the rarefied limits can be reliably obtained. To validate the accuracy and AP property of the GKLWS, several numerical benchmarks of isothermal low-speed rarefied gas flows are simulated by the present scheme. The numerical results show that the present scheme can be a reliable multiscale method for all Knudsen number low-speed isothermal gas flows.

Lax-Wendroff method for incompressible flow

The Lax-Wendroff scheme is extended to incompressible fluid flow problems within the framework of artificial compressibility method (ACM) utilizing a “cell-centered finite-volume” Δ-approximation on a non-orthogonal non-staggered grid. An ACM induces a transformation between conservative and primitive variables; the physical relevance of ACM is to bring about “matrix preconditionings”, provoking density perturbations. The coupled algorithm is pressure-based; it benefits from enhanced accuracy and greater flexibility using the “monotone upstream-centered schemes for conservation laws (MUSCL)” approach. Numerical experiments in reference to “buoyancy-driven” flows with strong sources illustrate that the entire strategy augments overall damping capability and robustness adhering to the factored pseudo-time integration method. Conclusively, the associated limiter function has little influence on the high resolution for selected test cases.

Lax–Wendroff Schemes with Polynomial Extrapolation and Simplified Lax–Wendroff Schemes for Dispersive Waves: A Comparative Study

One of the features of Boussinesq-type models for dispersive wave propagation is the presence of mixed spatial/temporal derivatives in the partial differential system. This is a critical point in the design of the time marching strategy, as the cost of inverting the algebraic equations arising from the discretization of these mixed terms may result in a nonnegligible overhead. In this paper, we propose novel approaches based on the classical Lax–Wendroff (LW) strategy to achieve single-step high-order schemes in time. To reduce the cost of evaluating the complex correction terms arising in the Lax–Wendroff procedure for Boussinesq equations, we propose several simplified strategies which allow to reduce the computational time at fixed accuracy. To evaluate these qualities, we perform a spectral analysis to assess the dispersion and damping error. We then evaluate the schemes on several benchmarks involving dispersive propagation over flat and nonflat bathymetries, and perform numerical grid convergence studies on two of them. Our results show a potential for a CPU reduction between 35 and 40% to obtain accuracy levels comparable to those of the classical RK3 method.

Numerical Solution of In-Viscid Burger Equation in the Application of Physical Phenomena: The Comparison between Three Numerical Methods

In this paper, upwind approach, Lax–Friedrichs, and Lax–Wendroff schemes are applied for working solution of In-thick Burger equation in the application of physical phenomena and comparing their error norms. First, the given solution sphere is discretized by using an invariant discretization grid point. Next, by using Taylor series expansion, we gain discretized nonlinear difference scheme of given model problem. By rearranging this scheme, we gain three proposed schemes. To verify validity and applicability of proposed techniques, one model illustration with subordinated to three different original conditions that satisfy entropy condition are considered, and solved it at each specific interior grid points of solution interval, by applying all of the techniques. The stability and convergent analysis of present three techniques are also worked by supporting both theoretical and numerical fine statements. The accuracy of present techniques has been measured in the sense of average absolute error, root mean square error, and maximum absolute error norms. Comparisons of numerical gets crimes attained by these three methods are presented in table. Physical behaviors of numerical results are also presented in terms of graphs. As we can see from numerical results given in both tables and graphs, the approximate solution is good agreement with exact solutions. Therefore, the present systems approaches are relatively effective and virtually well suited to approximate the solution of in-viscous Burger equation.

Investigation of Water Pollution in the River with Second-Order Explicit Finite Difference Scheme of Advection-Diffusion Equation and First-Order Explicit Finite Difference Scheme of Advection-Diffusion Equation

The perseverance of this research article is to investigate water pollution in rivers with a second-order explicit finite difference scheme of advection-diffusion equation (ADE) and a first-order explicit finite difference scheme of ADE. For investigation, two numerical schemes exploit here FTCSCS and second-order Lax-Wendroff type of ADE which is our new proposed one. In earlier Lax-Wendroff, type scheme existed only for hyperbolic partial differential equation (PDE), here a new second-order Lax-Wendroff type scheme is proposed for parabolic PDE and in addition assist to investigate water pollution with an expectation of better yield compared to the existing one. We implement numerical schemes to estimate the pollutant in water at different times and different points of water bodies. We investigate the numerical behaviour of water pollution by implementing the explicit centred difference scheme (FTCSCS) for advection-diffusion and for our proposed second-order Lax-Wendroff type scheme. Our computational result verifies the qualitative behavior of the solution of ADE for various considerations of the parameters.

Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation

In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.

A Numerical Model with Finite Difference Schemes for Multi-Species Solute Transport in Porous Media

A precise forecast of contaminant and solute transport has an inevitable role in the management of water resources. In accordance with this purpose, in this paper, a novel one-dimensional numerical model for the transport of a decay chain through homogeneous porous media is proposed. To develop the suggested model, two different schemes of the finite difference method, namely the Lax-Wendroff scheme and Fourth-Order scheme, are used. The verification and validation of the established model are examined by the analytical results of three multi-species solute dispersion problems with three- and four-chain members. The total mean square error, L2- and L∞-norms are applied to assess the results. Although analyses show that both schemes provided reliable results, the numerical results of the Lax-Wendroff scheme are more accurate.

Analysis of nodalization uncertainty for nuclear system analysis code with Lax-Wendroff numerical scheme

In modeling the complex systems such as a nuclear power plant (NPP), the nodalization becomes more important to satisfy Courant number limitation and reduce the numerical diffusion due to the 1st order numerical scheme. Since the higher-order numerical scheme can reduce the numerical diffusion problem, the accuracy can be improved and the nodalization uncertainty can be reduced by improving the current numerical scheme. However, another sources for the user effect uncertainty originates from a nodalization process and this can be affected by the selected numerical scheme. Sufficient sensitivity analyses should be performed on the nodalization to ensure that the calculated results are free from these uncertainties. Therefore, in this study, the nodalization uncertainties of 1st order upwind scheme and Lax-Wendroff scheme are evaluated with the revised MARS-KS code. The analysis of the nodalization uncertainty is performed for the separate effect tests (SETs) such as subcooled boiling and reflood experiments. The nodalization uncertainty is analyzed with the error between the experimental data and the calculation results of 1st order upwind scheme or Lax-Wendroff scheme. Thus, when using Lax-Wendroff scheme, the more accurate results with decreased nodalization uncertainty can be obtained. For the reflood calculations, accurate and consistent prediction of maximum peak cladding temperature (PCT) and accurate prediction of quenching time were observed with Lax-Wendroff scheme. However, the quenching time may not be consistent depending on the user. In terms of the computational efficiency, the central processing unit (CPU) time is generally increased by 40% with Lax-Wendroff scheme.