Lax Wendroff Research Articles

Finite volume transport schemes

We analyze arbitrary order linear finite volume transport schemes and show asymptotic stability in L 1 and L ∞ for odd order schemes in dimension one. It gives sharp fractional order estimates of convergence for BV solutions. It shows odd order finite volume advection schemes are better than even order finite volume schemes. Therefore the Gibbs phenomena is controlled for odd order finite volume schemes. Numerical experiments sustain the theoretical analysis. In particular the oscillations of the Lax–Wendroff scheme for small Courant numbers are correlated with its non stability in L 1. A scheme of order three is proved to be stable in L 1 and L ∞.

Two-dimensional lattice Boltzmann model for compressible flows with high Mach number

In this paper we present an improved lattice Boltzmann model for compressible Navier–Stokes system with high Mach number. The model is composed of three components: (i) the discrete-velocity-model by M. Watari and M. Tsutahara [Phys. Rev. E 67 (2003) 036306], (ii) a modified Lax–Wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included, (iii) artificial viscosity. The improved model is convenient to compromise the high accuracy and stability. The included dispersion term can effectively reduce the numerical oscillation at discontinuity. The added artificial viscosity helps the scheme to satisfy the von Neumann stability condition. Shock tubes and shock reflections are used to validate the new scheme. In our numerical tests the Mach numbers are successfully increased up to 20 or higher. The flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems.

A numerical scheme for morphological bed level calculations

The typical equation for bed level change in sediment transport in river, estuary and near shore systems is based on conservation of sediment mass. It is generally a nonlinear conservation equation for bed level. The physics here are similar to shallow water wave equations and gas dynamics equation which will develop shock waves in many circumstances. Many state-of-art morphological models use classical lower order Lax–Wendroff or modified Lax–Wendroff schemes for morphology which are not very stable for long time sediment transport processes simulation. Filtering or artificial diffusion are often added to achieve stability. In this paper, several shock capturing schemes are discussed for simulating bed level change with different accuracy and stability behaviors. The conclusion is in favor of a fifth order Euler-WENO scheme which is introduced to sediment transport simulations here over other schemes. The Euler-WENO scheme is shown to have significant advantages over schemes with artificial viscosity and filtering processes, hence is highly recommended especially for phase-resolving sediment transport models.

Control of numerical effects of dispersion and dissipation in numerical schemes for efficient shock-capturing through an optimal Courant number

Composite schemes consist of several steps of a dispersive scheme followed by one step of a dissipative scheme [Liska Richard, Wendroff Burton. Composite schemes for conservation laws. SIAM J Numer Anal 1998;35(6):2250–71]. The latter [Liska Richard, Wendroff Burton. 2D shallow water equations by composite schemes. Int J Numer Meth Fluids 1999;30:461–79] acts as a filter reducing oscillations in regions of discontinuity. Liska and Wendroff have derived the composite Lax-Wendroff/Lax-Friedrichs (LWLF) [Liska Richard, Wendroff Burton, 1998] scheme which blends the Lax-Wendroff (LW) scheme with the 2-step Lax-Friedrichs (LF) scheme. The formulation of the 2-step Lax-Friedrichs scheme [Liska Richard, Wendroff Burton, 1998] is different from that of the classic Lax-Friedrichs scheme and has been devised by Liska [Liska Richard, Wendroff Burton, 1998]. In this work, we propose to replace LW scheme by MacCormack (MC) scheme since the latter is less dispersive. We obtain a new composite scheme in 1-D and in 2-D by blending the MacCormack scheme with the 2-step Lax-Friedrichs scheme which we term as the composite MacCormack/Lax-Friedrichs (MCLF) scheme. This is followed by analytical work on the effective amplification factor (EAF) and the relative phase error (RPE) for both families of schemes in 1-D and 2-D: LWLFn and MCLFn, consisting of ( n − 1) steps of the dispersive scheme (LW or MC) and 1 step of the dissipative LF scheme. We introduce a new concept, baptised as Curbing of Dispersion by Dissipation for Efficient Shock-capturing, CDDES in which a cfl number is computed whereby dissipation curbs dispersion. This cfl number is termed as optimal in this work. We conduct a comparative study based on numerical experiments in 2-D namely: contact-discontinuity problem [Ould Kaber SM. A legendre pseudospectral viscosity method. J Comput Phys 1996;128:165–80], rotating hill problem [Ould Kaber SM, 1996] and the deformative flow of Smolarkiewicz [Dabdub Donald, Seinfeld John H. Numerical advective schemes used in air quality models-sequential and parallel implementation. Atmos Environ 1994;28(20):3369–85, Ghods A, Sobouti F, Arkani-Hamed J. An improved second order method for solution of pure advection problems. Int J Numer Meth Fluids 2000;32:959–77, Nguyen K, Dabdub D. Two-level time-marching scheme using splines for solving the advection equation. Atmos Environ 2001;35:1627–37] to show that the MacCormack/Lax-Friedrichs (MCLF) scheme is more efficient than LWLF scheme to capture shocks in regions of discontinuity. We also show that better results are obtained at optimal cfl numbers for some variants of LWLFn and MCLFn schemes, with n = 2, 3, 4 and 5.

Short-pulsed laser transport in absorbing and scattering media: time-based versus frequency-based approaches

Optical tomography (OT) is a promising non-intrusive characterization technique of absorbing and scattering media that uses transmitted and/or reflected signals of samples irradiated with visible or near-infrared light. The quality of OT techniques is directly related to the accuracy of their forward models due to the use of inversion algorithms. In this paper, forward models for transient OT approaches are investigated. The system under study involves a one-dimensional absorbing and scattering medium illuminated by a short laser pulse; this problem is solved using a discrete ordinates–finite volume (DO–FV) method in both time and frequency domain. Previous works have shown that time-domain approaches coupled with first order spatial interpolation schemes cannot represent the physics of the problem adequately as transmitted fluxes emerge before the minimal physical time required to leave the medium. In this work, the Van Leer and Superbee flux limiters, combined with the second order Lax–Wendroff scheme, are used in an attempt to prevent this. Results show that despite significant improvement, flux limiters fail to completely eliminate the physically unrealistic behaviour. On the other hand, results for transmittance obtained from the frequency-based method are accurate, without physically unrealistic behaviours at early time periods. The frequency-dependent approach is however computationally expensive, since it requires approximately five times more computational time than its temporal counterpart when used as a forward model for transient OT. On the other hand, the great advantages of the frequency-based approach is that limited windows of temporal signals can be calculated efficiently (in transient OT), and it can also be used as a forward model for steady-state, frequency-based and transient OT techniques.

Sub-Alfvenic inlet boundary conditions for axisymmetric MHD nozzles

There are numerous electromagnetic accelerator concepts which require plasma expansion through a magnetic nozzle. If the inlet flow is slower than one or all of the outgoing characteristics, namely, the Alfven, slow and fast magnetosonic speeds, then the number of inlet conditions which could be arbitrarily specified are reduced by the number of outgoing characteristics (up to three). We derive the axisymmetric compatibility equations using the method of projected characteristics for the inlet conditions in the z-plane to assure the boundary conditions being consistent with flow properties. We make simplifications to the equations assuming that the inlet Alfven speed is much faster than the sonic and slow magnetosonic speeds. We compare results for various inlet boundary conditions, including a modified Lax–Wendroff implementation of the compatibility equations, first order extrapolation and arbitrarily specifying the inlet conditions, in order to assess the stability and accuracy of various approaches.

An accurate time integration method for simplified overland flow models

An accurate time integration method for the diffusion-wave and kinematic-wave approximated models for the overland flow is proposed. The discretization of the first- and second-order spatial derivatives in the basic equation is obtained by using the second-order Lax–Wendroff and the three-point centred finite difference schemes, respectively. For the solution in time, the system of ordinary differential equations, obtained by the linearization of the celerity and of the hydraulic diffusivity by Taylor series expansions, is integrated analytically. The stability and the numerical dissipation and dispersion are investigated by the Fourier analysis. A proper Courant number, and the corresponding time step for the numerical simulations can be established. In addition, the proposed diffusion-wave and kinematic-wave models are straightforwardly extended to the two-dimensional flow. Test cases for both one- and two-dimensional problems, compare the solutions of the diffusion-wave and kinematic-wave models with analytical solutions, with experimental results and with numerical solutions obtained by the Saint–Venant equations. These simulations show that the proposed numerical–analytical models accurately predict the overland flow for several situations, in particular for unsteady rainfall rate and for spatial variations of the surface roughness.

Numerical predictions of short-pulsed laser transport in absorbing and scattering media. I—A time-based approach with flux limiters

A one-dimensional transient radiative transfer problem in the Cartesian coordinate system involving an absorbing and scattering medium illuminated by a short laser pulse has computationally been solved by use of a finite volume method (FVM). Previous works have shown that first order spatial interpolation schemes cannot represent the physics of the problem adequately as transmitted fluxes emerge before the minimal physical time required to leave the medium. In this paper, the Van Leer and Superbee flux limiters, combined with the second order Lax–Wendroff scheme, are used in an attempt to prevent this. Results presented in this work show that, despite significant improvement, flux limiters fail to completely eliminate the physically unrealistic behaviour. Therefore, a numerical approach into which temporal variables are transformed into frequency-dependent variables is presented in the companion paper to this work.

Analysis of the second vorticity confinement scheme

The second vorticity confinement scheme proposed by Steinhoff is analysed in detail. First, starting from the 1D linear transport equation applied to a pulse, various formulations for the confinement are compared. The linear 2nd-order schemes (Warming-Beam and Lax–Wendroff schemes) give oscillatory solutions, while the nonlinear confinements of same accuracy behave as limiters with artificial compression. Not only these 2nd-order schemes with confinement conserve the pulse as accurately as a 3rd-order one for identical and sufficiently fine grids, but they remain stable with a global negative diffusion which allows them to conserve the pulse endlessly concentrated over 5–8 mesh cells in the computation, although the form of the equations is then lowered to 1st-order. Application of the energy method for stability analysis indicates that the signal relaxes towards a constant energy solution, for which the energy brought in by the anti-diffusive confinement is balanced by the energy removed from the solution by the diffusive terms. Application to the Euler equations for the advection of a 2D vortex proves that a similar approach can be applied to nonlinear problems. The most appropriate formulation of compressible vorticity confinement is to apply it identically to the incompressible formulation, i.e. without source term in the energy conservation equation.

Time-domain computation of muffler frequency response: Comparison of different numerical schemes

A comparative study of the performance of different schemes used to solve one-dimensional gas flow equations when applied to the computation of the frequency response of exhaust mufflers is presented. Simple but representative systems with well-known acoustic behaviour were considered. Apart from the classical Lax–Wendroff and MacCormack schemes, the total variation diminishing schemes, flux corrected transport techniques or the innovative space–time conservation element and solution element method were considered. The results provide guidelines for a proper choice of the numerical scheme, taking into account the mesh spacing.

Nonlinear Uniaxial Thermoelastic Waves with Second Sound by Finite Difference–Flux Corrected Transport Method

Wave propagation in a nonlinear thermoelastic thin rod is studied by the Finite Difference–Flux-Corrected Transport (FD-FCT) method. In the Finite Difference method, Lax–Wendroff two-step scheme is used to discretize the basic equations. However, the Lax–Wendroff scheme itself cannot give good solutions especially at the wavefronts where the primary variables show strong jumps. The FCT scheme is added in order to reduce the oscillations at the wavefronts that are caused by the explicit finite difference method. In addition, a front tracking method is introduced to improve the solution when the jumps in variables across the wavefront are strong. The solutions obtained by the FD-FCT method along with the use of the front tracking approach are compared with those obtained by the method of characteristics. Copper and cast iron rods, which are subjected to two different types of boundary conditions, applied to one end of the rod are used in the computation and for comparison. “To know the mechanics of the wave is to know the entire secret of Nature” Walter Russell (www.svpvril.com).

Solution of population balances with growth and nucleation by high order moment-conserving method of classes

The high order method of classes, developed in our earlier work [Alopaeus, V., Laakkonen, M., Aittamaa J., 2006a. Solution of population balances with breakage and agglomeration by high order moment-conserving method of classes. Chemical Engineering Science 61, 6732–6752] for solution of population balances (PBs), is extended to problems with growth and primary nucleation. The growth problem leads to a hyperbolic partial differential equation with fundamentally different numerical characteristics than the PB with breakage and agglomeration only. However, we show that the principle of moment conservation in the numerical solution can also be applied to this advection-type problem, leading to extremely accurate numerical solutions. The method is tested for two numerical cases. The first one is mass transfer induced particle growth, and the second one is primary nucleation with constant growth (similar to the Riemann advection problem). For mass transfer induced growth, we first analyze functional form of the growth rate from mass transfer correlation viewpoint, and derive a general analytical solution for the power-law growth. The numerical results from the moment conserving method are also compared to one well established high resolution numerical method for advection problems, namely the Lax–Wendroff method with van Leer flux limiter. It was shown that the present method is far superior by predicting the distribution moments with several order of magnitudes lower numerical error. For the Riemann problem with constant growth rate, the present method predicts the shock front location exactly without any numerical diffusion.

An improved nearly analytical discrete method: an efficient tool to simulate the seismic response of 2-D porous structures

The nearly analytic discrete method (NADM) for acoustic and elastic waves in porous elastic media is a perturbation method recently proposed by Yang et al (2006a Commun. Comput. Phys. 1 528–47). This method uses the truncated Taylor series expansion to approximate the time derivatives and the local high-order interpolation to approximate the spatial high-order derivatives by simultaneously using the displacements and its gradients and the velocity. As a result, it can suppress effectively numerical dispersions caused by the discretizing the wave equations when too-coarse grids are used. In this paper, we present an improved nearly-analytic discrete method (INADM) for the porous case. We compare numerically the error of the INADM with those of the original NADM and the so-called Lax–Wendroff correction (LWC) schemes for 1-D and 2-D cases, and give the wave-field modelling in 2-D porous isotropic and anisotropic media. We show that, compared with the original NADM, the INADM for the 2-D case can reduce significantly the storage space and increase time accuracy, while the space accuracy remains the same as that of the original one. Numerical experiments show that the error of the INADM for the porous case is less than those of the NADM and the fourth-order LWC scheme. The three-component seismic wave-fields in the 2-D porous isotropic medium are compared with those obtained by using the NADM, the LWC method, and exact solutions. Several characteristics of waves propagating in porous anisotropic media, computed by the INADM, are also reported in this study. Promising numerical results illustrate that the INADM provides a useful tool for large-scale porous problems and it can effectively suppress numerical dispersions.

Numerical solutions of Burgers’ equation with random initial conditions using the Wiener chaos expansion and the Lax–Wendroff scheme

The work is concerned with efficient computation of statistical moments of solutions to Burgers’ equation with random initial conditions. When the Lax–Wendroff scheme is expanded using the Wiener chaos expansion (WCE), it introduces an infinite system of deterministic equations with respect to non-random Hermite–Fourier coefficients. One of important properties of the system is that all the statistical moments of the solution can be computed using simple formulae that involve only the solution of the system. The stability, accuracy, and efficiency of the WCE approach to computing statistical moments have been numerically tested and compared to those for the Monte Carlo (MC) method. Strong evidence has been given that the WCE approach is as accurate as but substantially faster than the MC method, at least for certain classes of initial conditions.

Dislocation transport using an explicit Galerkin/least-squares formulation

An explicit Galerkin/least-squares formulation is introduced for a quasilinear transport equation in field dislocation mechanics (FDM) and applied to the study of the kinematics of dislocation density evolution in the following physical contexts: annihilation of dislocations, expansion of a polygonal dislocation loop and simulation of a Frank–Read source. Stability analysis is carried out for the corresponding linear one-dimensional (1D) case. The formulation reduces to the Lax–Wendroff finite difference scheme for the 1D equation when equal weighting is used for the Galerkin and least-squares terms and the shape functions are linear. This conditionally stable method leads to a symmetric well-conditioned system of equations with constant coefficients, making it attractive for large-scale problems.It is shown that the transport equation, in the contexts mentioned above simplifies to the Hamilton–Jacobi equations governing geometrical optics and level-set methods. The weak solutions to these equations are not unique, and the numerical method is able to capture solutions corresponding to shock as well as rarefraction waves by appropriate algorithmic modifications.

A Numerical Comparison of the Lax–Wendroff Discontinuous Galerkin Method Based on Different Numerical Fluxes

Discontinuous Galerkin (DG) method is a spatial discretization procedure, employing useful features from high-resolution finite volume schemes, such as the exact or approximate Riemann solvers serv...

Explicit solutions to a convection-reaction equation and defects of numerical schemes

We develop a theoretical tool to examine the properties of numerical schemes for advection equations. To magnify the defects of a scheme we consider a convection-reaction equation u t + ( | u | q / q ) x = u , u , x ∈ R , t ∈ R + , q > 1 . It is shown that, if a numerical scheme for the advection part is performed with a splitting method, the intrinsic properties of the scheme are magnified and observed easily. From this test we observe that numerical solutions based on the Lax–Friedrichs, the MacCormack and the Lax–Wendroff break down easily. These quite unexpected results indicate that certain undesirable defects of a scheme may grow and destroy the numerical solution completely and hence one need to pay extra caution to deal with reaction dominant systems. On the other hand, some other schemes including WENO, NT and Godunov are more stable and one can obtain more detailed features of them using the test. This phenomenon is also similarly observed under other methods for the reaction part.

An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - II. The three-dimensional isotropic case

SUMMARY We present a new numerical method to solve the heterogeneous elastic wave equations formulated as a linear hyperbolic system using first-order derivatives with arbitrary high-order accuracy in space and time on 3-D unstructured tetrahedral meshes. The method combines the Discontinuous Galerkin (DG) Finite Element (FE) method with the ADER approach using Arbitrary high-order DERivatives for flux calculation. In the DG framework, in contrast to classical FE methods, the numerical solution is approximated by piecewise polynomials which allow for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann-Problems can be applied as in the finite volume framework. To define a suitable flux over the element surfaces, we solve so-called Generalized Riemann-Problems (GRP) at the element interfaces. The GRP solution provides simultaneously a numerical flux function as well as a time-integration method. The main idea is a Taylor expansion in time in which all time-derivatives are replaced by space derivatives using the so-called Cauchy–Kovalewski or Lax–Wendroff procedure which makes extensive use of the governing PDE. The numerical solution can thus be advanced for one time step without intermediate stages as typical, for example, for classical Runge–Kutta time stepping schemes. Due to the ADER time-integration technique, the same approximation order in space and time is achieved automatically. Furthermore, the projection of the tetrahedral elements in physical space on to a canonical reference tetrahedron allows for an efficient implementation, as many computations of 3-D integrals can be carried out analytically beforehand. Based on a numerical convergence analysis, we demonstrate that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes and computational cost and storage space for a desired accuracy can be reduced by higher-order schemes. Moreover, due to the choice of the basis functions for the piecewise polynomial approximation, the new ADER–DG method shows spectral convergence on tetrahedral meshes. An application of the new method to a well-acknowledged test case and comparisons with analytical and reference solutions, obtained by different well-established methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER–DG approach for tetrahedral meshes provides a numerical technique to approach 3-D wave propagation problems in complex geometry with unforeseen accuracy.

Particle methods revisited: a class of high order finite-difference methods

We propose a new analysis of particle method with remeshing. We derive a class of high-order finite difference methods. Our analysis is completed by numerical comparisons with Lax–Wendroff schemes for the Burger equation. To cite this article: G.-H. Cottet, L. Weynans, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Optimal Nearly Analytic Discrete Approximation to the Scalar Wave Equation

Recently, we proposed the so-called optimal nearly analytic discrete method (onadm) for computing synthetic seismograms in acoustic and elastic wave problems (Yang et al 2004). In this article, we explore the theoretical properties of the onadm including the stability criteria of the onadm for solving 1D and 2D scalar wave equations, numerical dispersion, theoretical error, and computational efficiency when using the onadm to model the acoustic wave fields. For comparison in the 1D case, we also discuss numerical dispersions and stability criteria of the so- called Lax–Wendroff schemes with accuracy of O (Δ t 4 , Δ x 8 ) and O (Δ t 4 , Δ x 10 ) and the pseudospectral method (psm). We then apply the onadm to the heterogeneous case in synthetic seismograms. Promising numerical results illustrate that the onadm provides a useful tool for large-scale heterogeneous practical problems because it can effectively suppress numerical dispersions caused by discretizing the wave equations when too-coarse grids are used. Numerical modeling also indicates that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for decreasing the numerical dispersion and source-generated noise caused by the discretization of wave equations because wave- displacement gradients include important seismic information.