Lax Wendroff Research Articles

An novel finite difference dispersion error elimination mechanism in the Lax–Wendroff high‐order time discretization

AbstractTime domain finite difference methods have been widely used for wave‐equation modelling in exploration geophysics over many decades. When using time domain finite difference methods, it is desirable to use a larger time step so as to save numerical simulation time. The Lax–Wendroff method is one of the well‐known methods to allow larger time step without increasing the time grid dispersion. However, the Lax–Wendroff method suffers from more time consumption because there are more spatial derivatives required to be approximated by the finite difference operators. We propose a new finite difference scheme for the Lax–Wendroff method so as to reduce the numerical simulation time. Then we determine the finite difference operator coefficients and analyse the dispersion error of the proposed finite difference scheme for the Lax–Wendroff method. At last, we apply the proposed finite difference scheme for the Lax–Wendroff method to different velocity models. The numerical simulation results indicate that the proposed finite difference scheme for the Lax–Wendroff method can effectively suppress time grid dispersion and is more efficient compared to the traditional finite difference scheme for the Lax–Wendroff method.

Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: Quasi-conservative formulation with subcell finite volume corrections

We present a novel quasi-conservative arbitrary high order accurate ADER (Arbitrary-Derivative) discontinuous Galerkin method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a modified Lax–Wendroff theorem. Within this formalism, we also reformulate classical results concerning smooth solutions, contact discontinuities and moving interfaces. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax–Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics.To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude the presentation by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble for which we are able to avoid the development of any spurious oscillations.

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.

Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations

Abstract We solve two numerical experiments described by 2D nonconstant coefficient advection–diffusion equations with specified initial and boundary conditions. Three finite difference methods, namely Lax–Wendroff, Du-Fort–Frankel and a nonstandard finite difference scheme, are derived and used to solve the two problems, whereby only the first problem has an exact solution. Stability analysis is performed to obtain a range of values of the time step size at a fixed spatial step size. We obtain the rate of convergence in space when the three methods are used to solve Problem 1. Computational times of the three algorithms are computed for Problem 1. Results are displayed for the two problems using the three methods at times T = 1.0 T=1.0 and T = 5.0 T=5.0 . The main novelty is the stability analysis, which is not straightforward as we are working with numerical methods discretising 2D nonconstant coefficient advection–diffusion equation where many parameters are involved. The second highlight is to determine the most efficient scheme from the three methods. Third, there are very few published studies on analysis and use of numerical methods to solve nonconstant coefficient advection–diffusion equations, and this is one of the very few rare articles treating such topics.

Accelerating the simulation of kinetic shear Alfvén waves with a dynamical low-rank approximation

We propose a dynamical low-rank algorithm for a gyrokinetic model that is used to describe strongly magnetized plasmas. The low-rank approximation is based on a decomposition into variables parallel and perpendicular to the magnetic field, as suggested by the physics of the underlying problem. We show that the resulting scheme exactly recovers the dispersion relation even with rank 1. We then perform a simulation of kinetic shear Alfvén waves and show that using the proposed dynamical low-rank algorithm a drastic reduction (multiple orders of magnitude) in both computational time and memory consumption can be achieved. We also compare the performance of robust first and second-order projector splitting, BUG (also called unconventional), and augmented BUG integrators as well as a FFT-based spectral and Lax–Wendroff discretization.

Discussion on different numerical treatments on the loss of hyperbolicity for the two-layer shallow water system

This paper focus on the numerical approximation of two-layer shallow water system. First, a new approximation of the eigenvalues of the system is proposed, which satisfies some interesting properties. From this approximation, we give an accurate estimation of the non-hyperbolic region, which improves significantly the one computed with the classic eigenvalues approximation. In particular, we estimate both the lower and upper boundaries of the non-hyperbolic region. We also give a simple algorithm that allows us to compute bounds for the external eigenvalues, even when complex eigenvalues arise. We design an efficient FV solver depending on the hyperbolic nature of the system, that combines the PVM-2U-FL scheme and a new solver introduced in this work, named IFCP-FL, which degenerates to the Lax–Wendroff method in smooth areas. Different strategies for the numerical treatment for the loss of hyperbolicity are considered and discussed. Two of them are based on adding a friction term, depending on the classic or the new eigenvalues approximations, and the third one considers a simplified hyperbolic model in areas with complex eigenvalues. Some numerical tests are performed, including the case of two-layer fluids with different ratio of densities. The application to a two-layer model for submarine landslides is also considered. In the latter case, we show how the treatments based on adding friction are not appropriate for this kind of applications, whereas the treatment based on changing to a hyperbolic model produces much better results.

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.

Data-driven and echo state network-based prediction of wave propagation behavior in dam-break flood

Abstract The computational prediction of wave propagation in dam-break floods is a long-standing problem in hydrodynamics and hydrology. We show that a reservoir computing echo state network (RC-ESN) that is well-trained on a minimal amount of data can accurately predict the long-term dynamic behavior of a one-dimensional dam-break flood. We solve the de Saint-Venant equations for a one-dimensional dam-break flood scenario using the Lax–Wendroff numerical scheme and train the RC-ESN model. The results demonstrate that the RC-ESN model has good prediction ability, as it predicts wave propagation behavior 286 time-steps ahead with a root mean square error smaller than 0.01, outperforming the conventional long short-term memory (LSTM) model, which only predicts 81 time-steps ahead. We also provide a sensitivity analysis of prediction accuracy for RC-ESN's key parameters such as training set size, reservoir size, and spectral radius. Results indicate that the RC-ESN is less dependent on training set size, with a medium reservoir size of 1,200–2,600 sufficient. We confirm that the spectral radius has a complex influence on the prediction accuracy and currently recommend a smaller spectral radius. Even when the initial flow depth of the dam break is changed, the prediction horizon of RC-ESN remains greater than that of LSTM.

Oscillating Nonlinear Acoustic Waves in a Mooney–Rivlin Rod

Harmonic wave excitation in a semi-infinite incompressible hyperelastic 1D rod with the Mooney–Rivlin equation of state reveals the formation and propagation of the shock wave fronts arising between faster and slower moving parts of the initially harmonic wave. The observed shock wave fronts result in the collapse of the slower moving parts being absorbed by the faster parts; hence, to the attenuation of the kinetic and the elastic strain energy with the corresponding heat generation. Both geometrically and physically nonlinear equations of motion are solved by the explicit Lax–Wendroff numerical tine-integration scheme combined with the finite element approach for spatial discretization.

The splitting-based semi-implicit finite-difference schemes for simulation of blood flow in arteries

This paper is devoted to construction and analysis of splitting-based finite-difference schemes for simulation of blood flow in applications of one-dimensional models. Proposed schemes are constructed for inviscid and viscid models of blood. The proposed approach is based on the modification of averaged incompressibility condition, rewritten for the square root of a cross-sectional area. It is approximated by an absolutely stable scheme with a skew-symmetric operator. The constructed schemes can be classified as semi-implicit. The stability of proposed schemes is analyzed by the method of energy inequalities. Sufficient stability conditions are obtained. For the problems with analytical solutions, it is demonstrated that a second-order convergence rate is achieved in practice.The proposed schemes are compared with the widely used explicit finite-difference schemes, such as Lax–Wendroff, Lax–Friedrichs and McCormack schemes. The benchmark problems for simulation of flows in the following vascular systems are considered: human carotid artery, human aorta with bifurcation, vessel with bifurcation and absorbing outflow conditions, and networks with 31, 63, and 127 vessels. It is demonstrated that the computations based on the splitting-based semi-implicit schemes can be performed much faster than for the explicit schemes with close values of relative errors.

Novel algorithm for compressible Newtonian axisymmetric thermal flow

In this paper, we mainly focus on the development of the Taylor-Galerkin/Pressure-Correction method (TG/PC) method to solve the flow of a compressible Newtonian fluid under nonisothermal conditions. The process of development needs applying the two-step Lax–Wendroff method on the energy equation, after which two new steps for energy equations will be obtained within the TG/PC algorithm. In addition, for the density component, the Tait equation of state is adopted to relate density to pressure, which also yields a new step within the novel method to give the correction formula of compressible pressure. To perform the analysis of the new algorithm, Poiseuille flow through axisymmetric rectangular channel for the Newtonian thermal flow is utilized as a simple problem test. The effect of nondimensional factors such as Reynolds number (Re) and Prandtl number (Pr) is discussed in this study. The influence of thermal conductivity [Formula: see text] and viscosity [Formula: see text] on the solution components is presented as well. Also, comparison for both compressible and incompressible flows is provided in addition to a comparison between both cases’ convergence, as it turns out that the convergence of the incompressible case is better than the compressible case. All the results of the effects presented in this study agree well with the approved physical trend.

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.

On the stability of totally upwind schemes for the hyperbolic initial boundary value problem

Abstract In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss–Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss–Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam–Warming scheme together with the simplified inverse Lax–Wendroff procedure at the boundary.

Shock Wave Formation and Cloaking in Hyperelastic Rods

The analysis of propagating an initially harmonic acoustic pulse in a semi-infinite hyperelastic rod obeying the Yeoh strain energy potential reveals attenuation with distance of the wave amplitudes caused by the elastic energy dissipation due to forming and propagation of the shock wave fronts and heat production. The observed attenuation of harmonic waves results in a broadband cloaking of fairly remote regions. The analysis is based on solving a nonlinear equation of motion by an explicit Lax–Wendroff time-difference scheme combined with the finite element discretization in the spatial domain. The revealing phenomena are applicable to studies of acoustic wave propagation in various elastomeric rubberlike materials modeled by the Yeoh hyperelastic potentials.

The Green’s function of the Lax–Wendroff and Beam–Warming schemes

We prove a sharp uniform generalized Gaussian bound for the Green’s function of the Lax–Wendroff and Beam–Warming schemes. Our bound highlights the spatial region that leads to the well-known (rather weak) instability of these schemes in the maximum norm. We also recover uniform bounds in the maximum norm when these schemes are applied to initial data of bounded variation.

Positivity-Preserving Lax–Wendroff Discontinuous Galerkin Schemes for Quadrature-Based Moment-Closure Approximations of Kinetic Models

The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor–corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear algebraic system in this prediction is solved using a Picard iteration. The correction is a straightforward explicit update based on the time-integral of the evolution equation, where the space-time prediction replaces all instances of the exact solution. In the absence of limiters, the high-order scheme does not guarantee that solutions remain in the convex set over which HyQMOM is moment-realizable. To overcome this, we introduce novel limiters that rigorously guarantee that the computed solution does not leave the convex set of realizable solutions, thus guaranteeing the hyperbolicity of the system. We develop positivity-preserving limiters in both the prediction and correction steps and an oscillation limiter that damps unphysical oscillations near shocks. We also develop a novel extension of this scheme to include a BGK collision operator; the proposed method is shown to be asymptotic-preserving in the high-collision limit. The HyQMOM and the HyQMOM-BGK solvers are verified on several test cases, demonstrating high-order accuracy on smooth problems and shock-capturing capability on problems with shocks. The asymptotic-preserving property of the HyQMOM-BGK solver is also numerically verified.

Quantifying parameter ranges for high fidelity simulations for prescribed accuracy by Lax–Wendroff method

Explicit second order central difference (CD2) based Lax–Wendroff (LW) method is evaluated for large eddy simulation here, using global spectral analysis (GSA) for its ability to compute fluid flows up to a specified accuracy. While the LW method has been analyzed for numerical stability in the past, rigorous quantification for performing accurate simulations including error dynamics has not been reported before. This is addressed here by determining optimal numerical parameters for prescribed accuracies based on the GSA of 2D convection–diffusion equation (CDE), which is the main focus of the present research. First, two LW methods are reported based on the application of the method for convection and both convection and diffusion operators, respectively, with the latter also having third derivative dispersive and fourth order diffusion terms. Using GSA, the optimal scheme is determined by comparing the two schemes for accuracy. The accuracy of GSA for wave propagation is also demonstrated by the validation of the q-waves determined from the analysis with the numerical simulations of 2D CDE. Finally, motivated by a one-to-one correspondence of the Navier–Stokes equation with the linear CDE established in “Effects of numerical anti-diffusion in closed unsteady flows governed by two-dimensional Navier–Stokes equation- (Suman et al., 2020)”, optimal simulation parameters are determined for the accuracy in representing physical diffusion of the scheme and validated using simulations of the canonical lid-driven cavity problem for a sub- and a super-critical Reynolds number.

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.