Lax Scheme Research Articles

Method of calculation of the flow motion model in water intake facilities

One of the urgent problems of our time is the uneven distribution of the flow velocity in the forebay of the intake structure, which leads to the destruction of the flow structure due to the water level, the amount of turbidity in the water, the size of the fraction and the change in speed. As a result, turbidity can be observed in the forebay and water intake units, an increase in hydraulic resistance in the suction network, a decrease in pump fluid loss, the formation of cavitation and the destruction of internal parts. In the article, the forebay of an irrigation pumping station and in the intake section, a mathematical model of the dynamics of water distribution over the velocity field, longitudinal and transverse distribution based on the shape of the forebay was created and numerous results were obtained. Taking into account the above, according to the identified Lax scheme, the Eyler differential equation of flow motion was numerically solved, and the calculation of the longitudinal and transverse velocity distributions in the forebay and water intake was carried out by setting the boundary conditions for the state of the flow.

A discrete Darboux–Lax scheme for integrable difference equations

We propose a discrete Darboux–Lax scheme for deriving auto-Bäcklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler–Yamilov type system which is related to the nonlinear Schrödinger (NLS) equation [7]. In particular, we construct an auto-Bäcklund transformation for this discrete system, its superposition principle, and we employ them in the construction of the one- and two-soliton solutions of the Adler–Yamilov system. PACS numbers02.30.Ik, 02.90.+p, 03.65.Fd Mathematics subject classification 202037K60, 39A36, 35Q55, 16T25.

2D numerical simulation of urban dam break and its effect to building using lax scheme with numerical filter

Dam-break is one of the disasters that can occurred due to earthquake. The earthquake vibration may damage the dam construction and therefore causing a dam-break flow. The flow can cause severe destruction to the downstream urban area. Dam-break modeling offers a way to analyze its effect of buildings. In this study, a 2D model for analyzing dam break flow is developed based on the Saint Venant equations and solved using Lax Scheme. The initial condition of the modelling is a rectangular channel with obstacles at both sides of the channel and two similar columns in the middle of the channel. These obstacles are considered as buildings in an urban area. Numerical filter is used to increase the stability of the simulation. The developed model is able to perform well in simulating a case of urban dam-break based on a previous experiment. In addition, the applied numerical filter is able to handle shock, therefore maintaining the stability of the model while reducing the simulation time without the needs to use a higher order numerical scheme.

On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants

A general elliptic N × N matrix Lax scheme is presented, leading to two classes of elliptic lattice systems, one which we interpret as the higher-rank analogue of the Landau–Lifschitz equations, while the other class we characterize as the higher-rank analogue of the lattice Krichever–Novikov equation (or Adlerʼs lattice). We present the general scheme, but focus mainly on the latter type of models. In the case N = 2 we obtain a novel Lax representation of Adlerʼs elliptic lattice equation in its so-called 3-leg form. The case of rank N = 3 is analyzed using Cayleyʼs hyperdeterminant of format , yielding a multi-component system of coupled 3-leg quad-equations.

NUMERICAL MODEL TO SIMULATE THE EROSION ON THE SLOPE DUE TO OVERTOPPING

The numerical model is developed consisting of a 1D flow model and the morphological model to simulate the erosion due to the water overtopping. The step method is applied to solve the water surface on the slope and the finite difference method of the modified Lax Scheme is applied for bed change equation. The Meyer-Peter and Muller formulae is used to determine the bed load transport rate. The model is calibrated and verified based on the data in experiment. It is found that the computed results and experiment data are good agreement.

Generalized Euler method for Hamilton Jacobi differential functional systems

Nonlinear first order partial functional differential systems are considered in the paper. Classical solutions of the local Cauchy problem on the Haar pyramid are approximated by solutions of suitable quasilinear systems of difference functional equations. The proposed numerical methods are difference schemes of the Euler type. A complete convergence analysis is given and it is shown by examples that the new methods are considerable better than the classical methods. It is shown that the Lax scheme is superfluous for the numerical approximations of classical solutions to nonlinear functional differential problems. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators.

Multiple scattering by the iterative Foldy–Lax scheme

The Foldy-Lax multiple-scattering equations in matrix form were examined for their suitability for solving the phase function and degree of linear polarization of clusters of particles as a complementary approach to the superposition T-matrix scheme. The constituent particles used for verification against the latter method were spheres with size parameters close to unity, with their aggregation governed by generalized Brownian motion. The number of constituents can be enlarged beyond the limitations of the benchmark method by a relatively small sacrifice in accuracy, but increasing constituent size and introducing nonsphericity easily result in numerical noise and overwhelming computational effort unless a large amount of the accuracy is going to be traded off.

Nutation Damper Undergoing a Coupled Motion

A novel numerical model is proposed to simulate liquid sloshing in a rectangular nutation damper (i.e. a tuned liquid damper) undergoing a coupled horizontal and rotational motion. Shallow water theory is used consistently to derive the governing equations of motion so that the model is applicable to large sloshing involving a hydraulic jump. It can also accommodate exposure of part of the damper’s floor to air by using a somewhat improved boundary shear approximation. A simple finite difference approach – the Lax scheme – is found to solve the equations of motion surprisingly well. Numerical predictions are checked against limited experimental data for a purely horizontal motion. Good agreement is generally observed. Furthermore, to demonstrate the model’s broader scope, the effect of a rotation is also considered in conjunction with a horizontal motion. The rotation is shown to significantly enhance the damper’s energy dissipation and, hence, its attenuation capability. For convenient practical application, an equivalent singledegree-of-freedom oscillator model is presented to characterize a nutation damper’s behavior for a coupled motion. The equivalent parameters of the model are determined so that the dissipated energy “best” fits a numerical counterpart. Their effect is investigated for different lengths, depths, and vibration levels of the damper. While the motivation of this investigation is to control the wind-induced galloping of overhead power lines, the proposed approach is applicable more generally to any excitation that induces low frequency vibrations.

Summing Lax series for nonparaxial beam propagation.

It is shown that the approach proposed by Lax et al. [Phys. Rev. A 11, 1365 (1975)] for studying the propagation of an electromagnetic beam beyond the paraxial approximation can be efficiently employed evenwhen the beam under consideration presents a very nonparaxial character. The method that we present consists of applying a nonlinear resummation scheme, the so-called delta transformation, to the divergent perturbative series arising from the Lax scheme. Numerical results pertinent to the evaluation of transverse and longitudinal components of the electric field are presented for the particular case of vectorial Gaussian beams, showing the effectiveness of the method.

An Integer Lattice Realization of a Lax Scheme for Transport Processes in Multiple Component Fluid Flows

A Lax–Wendroff-like finite-difference representation for the transport of multiple chemical components is formulated via integer variables. This representation ensures exactly the desired conservation laws at all times and achieves low numerical diffusivity. The algorithm requires less memory as compared to its floating-point predecessor, hence much less than standard lattice gas and lattice Boltzmann methods to date. Analytical and numerical studies demonstrate that the algorithm is stable under subsonic conditions.

Numerical solution of transport equation for applications in environmental hydraulics and hydrology

The advective term in the one-dimensional transport equation, when numerically discretized, produces artificial diffusion. To minimize such artificial diffusion, which vanishes only for Courant number equal to unity, transport owing to advection has been modeled separately. The numerical solution of the advection equation for a Gaussian initial distribution is well established; however, large oscillations are observed when applied to an initial distribution with sleep gradients, such as trapezoidal distribution of a constituent or propagation of mass from a continuous input. In this study, the application of seven finite-difference schemes and one polynomial interpolation scheme is investigated to solve the transport equation for both Gaussian and non-Gaussian (trapezoidal) initial distributions. The results obtained from the numerical schemes are compared with the exact solutions. A constant advective velocity is assumed throughout the transport process. For a Gaussian distribution initial condition, all eight schemes give excellent results, except the Lax scheme which is diffusive. In application to the trapezoidal initial distribution, explicit finite-difference schemes prove to be superior to implicit finite-difference schemes because the latter produce large numerical oscillations near the steep gradients. The Warming-Kutler-Lomax (WKL) explicit scheme is found to be better among this group. The Hermite polynomial interpolation scheme yields the best result for a trapezoidal distribution among all eight schemes investigated. The second-order accurate schemes are sufficiently accurate for most practical problems, but the solution of unusual problems (concentration with steep gradient) requires the application of higher-order (e.g. third- and fourth-order) accurate schemes.

The Method of Space-Time Conservation Element and Solution Element—A New Approach for Solving the Navier-Stokes and Euler Equations

A new numerical framework for solving conservation laws is being developed. This new framework differs substantially in both concept and methodology from the well-established methods, i.e., finite difference, finite volume, finite element, and spectral methods. It is conceptually simple and designed to overcome several key limitations of the above traditional methods. A two-level scheme for solving the convection-diffusion equation ∂u/∂t + a ∂u/∂x - μ ∂2u/∂x2 = 0 (μ ≥ 0) is constructed and used to illuminate major differences between the present method and those mentioned above. This explicit scheme, referred to as the a-μ scheme, has two independent marching variables unj and (ux)nj which are the numerical analogues of u and ∂u/∂x at (j, n), respectively. The a-μ scheme has the unusual property that its stability is limited only by the CFL condition, i.e., it is independent of μ. Also it can be shown that the amplification factors of the a -μ scheme are identical to those of the Leapfrog scheme if μ = O, and to those of the DuFort-Frankel scheme if a = O. These coincidences are unexpected because the a-μ scheme and the above classical schemes are derived from completely different perspectives, and the a -μ scheme does not reduce to the above classical schemes in the limiting cases. The a-μ scheme is extended to solve the 1D time-dependent Navier-Stokes equations of a perfect gas. Stability of this explicit solver also is limited only by the CFL condition. In spite of the fact that it does not use (i) any techniques related to the high-resolution upwind methods, and (ii) any ad hoc parameter, the current Navier-Stokes solver is capable of generating highly accurate shock tube solutions. Particularly, for high-Reynolds-number flows, shock discontinuities can be resolved within one mesh interval. The inviscid (μ = 0) a-μ scheme is reversible in time. It also is neutrally stable, i.e., free from numerical dissipation. Such a scheme generally cannot be extended to solve the Euler equations. Thus, the inviscid version is modified. Stability of this modified scheme, referred to as the a-ε scheme, is limited by the CFL condition and 0 ≤ ε ≤ 1, where ε is a special parameter that controls numerical dissipation. Moreover, if ε = 0, the amplification factors of the a-ε scheme are identical to those of the Leapfrog scheme, which has no numerical dissipation. On the other hand, if ε = 1, the two amplification factors of the a-ε scheme become the same function of the Courant number and the phase angle. Unexpectedly, this function also is the amplification factor of the highly diffusive Lax scheme. Note that, because the Lax scheme is very diffusive and it uses a mesh that is staggered in time, a two-level scheme using such a mesh is often associated with a highly diffusive scheme. The a-ε scheme, which also uses a mesh staggering in time, demonstrates that it can also be a scheme with no numerical dissipation. The Euler extension of the a -ε scheme has stability conditions similar to those of the a -epsiv; scheme itself. It has the unusual property that numerical dissipation at all mesh points can be controlled by a set of local parameters, Moreover, it is capable of generating accurate shock tube solutions with the CFL number ranging from close to 1 to 0.022

A numerical simulation of viscoelastic fluid flow in a two-dimensional channel. Application of Lax's scheme to the constitutive equation.

A new method of numelical analysis of a viscoelastic fluid flow is presented. The present method is derived by applying Lax's scheme to the constitutive equation and makes it possible to deal with a flow at a high Weissenberg number and a finite Reynolds number. The present numerical results are compared with experimental observations of Cochrane et al. on a flow over a protuberance, and the numerical predictions of a vortex behind a protuberance agree with the experimental reaults. A planar contraction flow is simulated numerically by the present method and the behaviour of the secondary flow vortex is investigated in detail. As a result, three pattelns of entry flow are observed.

Dam‐break Flows in Curved Channel

Experimental data on dam break flows obtained on a test facility are presented in this paper. The test facility comprises a 3.65 m by 2.3 m upstream reservoir and a 0.3 by 0.3 m rectangular downstream channel. This channel has two straight segments that are connected by a 180° bend. Capacitance probes are used to record water levels in the upstream reservoir. A new procedure using electronic digitization of video images is used to record water levels in the downstream channel. This procedure does not disturb flow and gives accurate results. Comparison of the unsteady water levels computed by using the Lax scheme with the measured water levels is satisfactory. The conservation form of the St. Venant equations predicts the height and celerity of the wave better than the nonconservation form. An approximate procedure is presented to compute the lateral gradient of water surface in a curved channel. Comparison with the measured results shows satisfactory agreement.

Study of Green's difference function for a wave system

Estimates are obtained for Green's difference function and for Green's function of the first differential approximation of Lax's scheme for a wave system. Using them we find an estimate for the increase of the solution in the norms of L and estimates, correct in respect of order, for Green's difference function and Green's function of the first differential approximation in the norms of L P .