Lax Method Research Articles

A multivariate model of deep slurry infiltration in slurry shield tunneling

There are multiple factors that influence slurry infiltration processes, making it necessary to consider these factors when comprehensively analyze the process of deep slurry infiltration. Firstly, this paper proposes a theoretical model for multivariate deep slurry infiltration and introduces a control equation governing the variation of the slurry infiltration domain. This equation accurately reflects changes in the infiltration boundary during the slurry infiltration process and meticulously responds to variations in pore water pressure, slurry retention rate, and pressure conversion efficiency-critical factors in the slurry infiltration process. Then, the Lax method was used to solve the corresponding control equations of the model, realizing the effective solution of the implicit partial differential equation system. The computational results of the theoretical model were compared with the indoor test results. Finally, the validated model was used to analyze the deep slurry infiltration characteristics. The results indicate that the theoretical model can reasonably simulate the deep slurry infiltration process, and the use of dynamic computational domain to describe deep slurry infiltration process is more in line with engineering practice. Furthermore, this model has guiding significance for the division of deep slurry infiltration stage and slurry surface infiltration stage.

Second-Order Accuracy in Time of Finite Difference Methods for Computational Aeroacoustics

The recently developed second-order accuracy in time finite difference method suitable for computational aeroacoustics (CAA) is introduced. Although, it is straight forward to compute the coefficients for finite-difference method of any order of accuracy using the Taylor series and to then further optimize them to enhance their wavenumber preserving properties, there are difficult questions concerning their numerical stability The goal of this work is to develop an effective numerical technique that includes both linear and nonlinear wave propagation in order to solve acoustics problems in time and space. It also aims to evaluate the accuracy, effectiveness, and stability of the new technique. In 1-D linear and nonlinear computational aeroacoustics, the novel techniques were used. The findings of the conventional methods (square wave (FTCS) technique and step wave lax approach) are presented in this paper, and it is shown that the FTCS method is typically unstable for hyperbolic situations and cannot be employed. Unfortunately, the FTCS equation has very little practical application. It is an unstable method, which can be used only (if at all) to study waves for a short fraction of one oscillation period. Nonlinear instability and shock formation are thus somewhat controlled by numerical viscosity such as that discussed in connection with Lax method equation. The second-order accuracy in time finite difference method is more efficient than the (square wave (FTCS), step wave lax) methods and is faster than the step wave lax method.

Acoustic Analysis Using Symmetrised Implicit Midpoint Rule

In wave propagation phenomena, time-advancing numerical methods must accurately represent the amplitude and phase of the propagating waves. The acoustic waves are non-dispersive and non-dissipative. However, the standard schemes both retain dissipation and dispersion errors. Thus, this paper aims to analyse the dissipation, dispersion, accuracy, and stability of the Runge–Kutta method and derive a new scheme and algorithm that preserves the symmetry property. The symmetrised method is introduced in the time-of-finite-difference method for solving problems in aeroacoustics. More efficient programming for solving acoustic problems in time and space, i.e. the IMR method for solving acoustic problems, an advection equation, compares the square-wave and step-wave Lax methods with symmetrised IMR (one-and two-step active). The results of conventional methods are usually unstable for hyperbolic problems. The forward time central space square equation is an unstable method with minimal usefulness, which can only study waves for short fractions of one oscillation period. Therefore, nonlinear instability and shock formation are controlled by numerical viscosities such as those discussed with the Lax method equation. The one- and two-step active symmetrised IMR methods are more efficient than the wave method.

Numerical simulation study of sympathetic detonation in stages

A Direct Numerical Simulation (DNS) scheme for solving the temporal-spatial multiscale problem of sympathetic detonation is proposed. According to the physical and chemical properties, the SD process is divided into two stages in the numerical simulation. Two different grid sizes are used in the two stages to improve calculation accuracy and efficiency. The local Level Set Method is used to accurately track the multi-material interface, and the Harten Lax and van Leer Contact (HLLC) method is used to solve local Riemann problem. Based on the computing method mentioned above, choosing the high evolution Weighted Essentially Non-Oscillatory (WENO) scheme for the spatial discretization of governing equations in alliance with Total Variation Diminishing (TVD) Runge-Kutta for time discretization, the Eulerian code of SD is developed. It solves the data interaction between the two solvers with variable grid size and time step, and realizes the three-dimensional parallel computing of SD. Composition B is taken as the research object. The processes of SD in different stand-off distances are analyzed. The transmission and safety distances of Composition B are given. The test under the same conditions is designed to verify the numerical simulation. The numerical simulation results are in good agreement with the experiment, which proves the accuracy and feasibility of the numerical method.

Propagation of Waves in Randomly Distributed Cylinders Using Three-Dimensional Vector Cylindrical Wave Expansions in Foldy–Lax Equations

In this article, we develop a hybrid method to calculate the propagation of microwaves in randomly distributed dielectric cylinders. The hybrid method combines off-the-shelf techniques for single object and our developed techniques of Foldy–Lax (FL) method that include extracting the T-matrix for single object, vector translation addition theorem, and solving FL multiple scattering equations. For cylindrical scatterers such as tree trunks, the T-matrix in vector three-dimensional cylindrical waves are extracted from infinite cylinder approximation (ICA). In solving FL to calculate statistical moments, we iterate one order of multiple scattering at a time, with averaging over realizations performed after each order. This physically based iterative method of calculating statistical moments converges faster than the traditional iterative method of calculating the exact solution for each realization. The main purpose is to simulate tall tree trunks at the L-band and ICA is of sufficient accuracies. Numerical results are illustrated for a large number of cylinders of up to 196 and cylinder lengths of up to 94 wavelengths, which are typical of forests at the L-band. Results of the simulations of the hybrid method show that the transmission coefficients of waves are several times larger than that of the commonly used models of the radiative transfer equation and distorted Born approximation.

Solving a mixture model of two-phase flow with velocity non-equilibrium using WENO wavelet methods

PurposeThe purpose of this work is to present the implementation of weighted essentially non-oscillatory (WENO) wavelet methods for solving multiphase flow problems. The particular interest is gas–liquid two-phase mixture with velocity non-equilibrium. Numerical simulations are carried out on different scenarios of one-dimensional Riemann problems for gas–liquid flows. Results are validated and qualitatively compared with solutions provided by other standard numerical methods.Design/methodology/approachThis paper extends the framework of WENO wavelet adaptive method to a fully hyperbolic two-phase flow model in a conservative form. The grid adaptivity in each time step is provided by the application of a thresholded interpolating wavelet transform. This facilitates the construction of a small yet effective sparse point representation of the solution. The method of Lax–Friedrich flux splitting is used to resolve the spatial operator in which the flux derivatives are approximated by the WENO scheme.FindingsHyperbolic models of two-phase flow in conservative form are efficiently solved, as shocks and rarefaction waves are precisely captured by the chosen methodology. Substantial computational gains are obtained through the grid reduction feature while maintaining the quality of the solutions. The results indicate that WENO wavelet methods are robust and sufficient to accurately simulate gas–liquid mixtures.Originality/valueResolution of two-phase flows is rarely studied using WENO wavelet methods. It is the first time such a study on the relative velocity is reported in two-phase flows using such methods.

Twisting Anderson pseudospins with light: Quench dynamics in terahertz-pumped BCS superconductors

We study the preparation (pump) and the detection (probe) of far-from-equilibrium BCS superconductor dynamics in THz pump-probe experiments. In a recent experiment [R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Phys. Rev. Lett. {\bf 111}, 057002 (2013)], an intense monocycle THz pulse with center frequency $\omega \simeq \Delta$ was injected into a superconductor with BCS gap $\Delta$; the subsequent post-pump evolution was detected via the optical conductivity. It was argued that nonlinear coupling of the pump to the Anderson pseudospins of the superconductor induces coherent dynamics of the Higgs (amplitude) mode $\Delta(t)$. We validate this picture in a two-dimensional BCS model with a combination of exact numerics and the Lax reduction method, and we compute the nonequilibrium phase diagram as a function of the pump intensity. The main effect of the pump is to scramble the orientations of Anderson pseudospins along the Fermi surface by twisting them in the $xy$-plane. We show that more intense pump pulses can induce a far-from-equilibrium phase of gapless superconductivity ("phase I"), originally predicted in the context of interaction quenches in ultracold atoms. We show that the THz pump method can reach phase I at much lower energy densities than an interaction quench, and we demonstrate that Lax reduction (tied to the integrability of the BCS Hamiltonian) provides a general quantitative tool for computing coherent BCS dynamics. We also calculate the Mattis-Bardeen optical conductivity for the nonequilibrium states discussed here.

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016

The choice of Numerical Integration Methods for the Subsystems with the Distributed Parameters

This article deals with methods of numerical modeling of heterogeneous electric circuits with distributed parameters for their possible use in the method of partitioning integration. We propose the use of modified Lax (LW) method for calculating the state of components with distributed parameters at the transient simulation in high-Q electric circuits, for example a transmission line. A comparison of transient simulation results in such circuits with the simulation results by known software systems is presented.

Coupled dam-break flow and bed load modelling using HLLC-WAF scheme.

A two-dimensional numerical model predicting flow over a mobile bed has been developed. Governing equations consist of the shallow water equations and the Exner equation. The finite volume method on an unstructured triangular grid was deployed to discretize the governing equations. The local Riemann problem is solved by the Harten, Lax and van Leer-contact (HLLC) method in the interface of the cells and the equations are solved using a fully coupled method. Then the flux modelling has been deployed by the total variation diminishing (TVD) version of the weighted average flux (WAF) scheme. The model was verified by comparison of the results and available experimental data for dam-break flow, in a laboratory test, via a channel with sudden enlargement and erodible bed conditions. Comparison of these two sets of results shows that increasing the accuracy of flux modelling caused the model results to have a reasonable agreement with the experimental data.

Application of the two-step Lax and Wendroff FCT and the CE-SE method to flow transport in wall-flow monoliths

Gas dynamic codes are computational tools applied to the analysis of air management in internal combustion engines. The governing equations in one-dimensional elements are approached assuming compressible unsteady non-homentropic flow and are commonly solved applying finite difference numerical methods. These techniques can also be applied to the calculation of flow transport in complex systems such as wall-flow monoliths. These elements are characterized by alternatively plugged channels with porous walls. It filters the particulates when the flow goes through the wall from the inlet to the outlet channels. Therefore, this process couples the solution of every pair of inlet and outlet channels. In this study, the adaptation of the two-step Lax and Wendroff method and the space-time Conservation Element and Solution Element method is performed to be applied in the solution of flow transport in wall-flow monolith channels. The influence on the prediction ability is analysed by a shock-tube test and experimental data obtained under impulsive flow conditions.

EXISTENCE AND UNIQUENESS OF LAX-TYPE SOLUTIONS TO THE RIEMANN PROBLEM OF SCALAR BALANCE LAW WITH SINGULAR SOURCE TERM

We give a new approach of constructing the generalized entropy solutions to the Riemann problem of scalar nonlinear balance laws with singular source terms. The source term is singular in the sense that it is a product of delta function and a discontinuous function, which is undefined in distribution. By re-formulating the source term, we study the corresponding perturbed Riemann problem. The existence and stability of perturbed Riemann solutions is established under some entropy condition so that the generalized entropy solutions of Riemann problem can be interpreted as the limit of corresponding perturbed Riemann solutions. The self-similarity of generalized entropy solutions is also obtained, which means that Lax's method in [13] can be extended to scalar nonlinear balance laws with singular source terms.

Higher order analogues of Tracy–Widom distributions via the Lax method

We study the distribution of the largest eigenvalue in Hermitian one-matrix models when the spectral density acquires an extra number of k − 1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy nonlinear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990s. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painlevé XXXIV hierarchy. These are the higher order analogues of the Tracy–Widom distribution. Using known Bäcklund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painlevé II hierarchy.

A riemann-hilbert problem in a riemann surface

One of the inspirations behind Peter Lax's interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann's conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs.Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an “equilibrium measure” problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the complex plane) could apply to previously intractable problems and also produce more detailed information.Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solutions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background.Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift-Zhou “nonlinear stationary phase method” to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!

A Two-Scale Multiple Scattering Problem

Consider the scattering problem of a time-harmonic plane wave incident on a heterogeneous medium consisting of isotropic point (small scale) scatterers and an extended (wavelength comparable) obstacle scatterer in three-dimensional space. To compute the scattered field from the interaction between the incident wave and the point scatterers only, the Foldy–Lax method provides an effective approach, while boundary integral equation methods play an important role for solving the scattering problem solely involving an extended obstacle scatterer. It is a challenging two-scale multiple scattering problem when both the point scatterers and the extended obstacle are present. In this paper, a generalized Foldy–Lax method is developed to fully take account of the multiple scattering in the heterogenous medium. The method is viewed from two different formulations: the series solution and the integral equation. The series solution formulation is shown as an efficient iterative scheme to the integral equation formulation. The convergence of the scattered fields and the far-field patterns from the series solution formulation are characterized in terms of scattering coefficients. Numerical experiments are presented to show the agreement and the effectiveness of the proposed two approaches.

Global solutions for initial–boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial–boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial–boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.

Globally Lipschitz continuous solutions to a class of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax's method and generalized Glimm's method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.

Theory of optical spectra of impurity centres in crystals: general consideration of quadratic vibronic coupling

A method is proposed for the calculation of the spectra of the multiphonon transitions caused by the quadratic vibronic coupling with a phonon continuum. In the method, the time evolution of the final state is described by the path integrals. By applying the Stratonovich–Hubbard identity, the quadratic coupling is presented as the fluctuating linear coupling; the latter is calculated by the Lax method. As a result, the problem is reduced to the calculation of the determinants of matrices, the elements of which are given by the pair correlation functions of the contributing configurational coordinates. The method has been verified for the case of two mixed modes. Numerical calculations have also been made for a modified Debye model.

Two-layer shallow water computation of mud flow intrusions into quiescent water

The present computational study addresses two-layer shallow flows in which the superposed layers differ in velocity, density and rheology. The geomorphological phenomena motivating this model are confluence problems in which mud and debris surges slump into upland lakes and rivers. Specifically, the flows of interest are assumed to be sharply stratified, with a clear water layer flowing over a moving layer of mud modelled as a Herschel–Bulkley fluid.A finite volume computational scheme suitable for the simulation of such flows is presented and applied to various validation cases. The scheme extends to two-layer flows the robust method of Harten, Lax and Van Leer. Special care is devoted to the following numerical issues: the treatment of pressure along interfaces, the ”contact points“ at which the thickness of either layer can vanish, and the finite yield stress characterizing the mud rheology. Results are presented for the intrusion of mud surges into shallow quiescent water.

Error estimation of numerical solutions of linear convection–diffusion problem

We present a new error estimating method of numerical solutions using the sensitivity analysis and modified equation techniques. The method can investigate not only the effects of numerical error but also those of uncertainty in a physical model at the same time. In this paper, we apply it to the finite-difference method of upwind and Lax–Wendroff schemes for the linear convection–diffusion problems. If a standard case of typical parameters is computed with the method, no additional computation is required to estimate the other numerical parameters' results such as more detailed solutions. Furthermore, we can quantitatively estimate the numerical error only from the sensitivity analysis results.