Lax-Friedrichs Scheme Research Articles

Magnetohydrodynamic Modeling of Background Solar Wind near Mars: Comparison with MAVEN and Tianwen-1

Combined with data assimilation methods, a three-dimensional magnetohydrodynamic (MHD) numerical model is an effective tool to explore the mechanism of space weather. As a driver of space weather, the dynamic development of stream interaction regions (SIRs) near the orbit of Mars is an area of active research. In this study, we use the interplanetary total variation diminishing (TVD) MHD model to simulate solar wind parameters and model SIRs near Mars from 2021 November 15 to 2021 December 31. In this model, the MHD equations are solved by the conservation TVD Lax–Friedrichs scheme in a rotating spherical coordinate system with six component meshes used on the spherical shell. Solar wind velocity, density, temperature, and magnetic field strength are given at the inner boundary due to the characteristic waves propagating outward. We compared modeled results with observations from Mars Atmospheric Volatile EvolutioN (MAVEN) and Tianwen-1 (China’s first Mars exploration mission). Statistical analysis shows that the simulated results can capture SIRs and are in good agreement with observations; moreover, the assimilated results based on the Kalman filter improve the accuracy of numerical prediction compared with simulated results. This paper is the first attempt to simulate SIR events combined with MAVEN and Tianwen-1 in situ observations. Our work demonstrates that using the MHD model with the Kalman filter to reconstruct solar wind parameters can help us study the characteristics of SIRs near Mars, improve the capabilities of space weather forecasting, and understand the background solar wind environment.

Riemann problem for a general variable coefficient Burgers equation with time-dependent damping

In this paper, we study the Riemann problem for a general variable coefficient Burgers equation with time-dependent damping. We use a nonlinear time-dependent viscosity equation with a similarity variable. Thus, when the viscosity goes to zero, we obtain Riemann solutions to the general variable coefficient Burgers equation with time-dependent damping. Moreover, we use the Lax–Friedrichs scheme to obtain numerical evidence of the Riemann solutions.

Numerical study for two models in chromatography using modified Rusanov scheme

We investigate the conservation laws that explain the chemistry process of isolating a single component from a mixture using column chromatography. We construct the modified Rusanov (mR) technique to solve the chromatography system. This technique is divided into two parts, the first of which is reliant on a local parameter which enables diffusion control. The conservation equation is recovered in stage two. It is crucial to make sure the numerical approach is built in a way that it is conservative on the overlap and, additionally, that the approximative solutions converge to the weak solution. This strategy is clear-cut, not difficult to use and precise. We compare the mR scheme via the Rusanov scheme and Lax–Friedrichs scheme via five test cases. This numerical research validates the mR scheme's efficacy. Finally, the mR approach can be utilized to solve a variety of conservation laws arising in natural science.

Compactness estimates for difference schemes for conservation laws with discontinuous flux

Abstract We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t)$ is $BV$-regular and may exhibit discontinuities along curves in the $(x,t)$ plane. Our approach, which is technically elementary, relies on a discrete interaction estimate and one entropy function. While the details are specifically outlined for the Lax-Friedrichs scheme, the same framework can be applied to other difference schemes. Notably, our compactness estimates are new even in the homogeneous case ($k\equiv 1$).

Computational uncertainty and optimal grid size and time step of the Lax–Friedrichs scheme for the 1D advection equation

This paper examines truncation and round-off errors in the numerical solution of the 1D advection equation with the Lax–Friedrichs scheme, and accumulation of the errors as they are propagated to high temporal layers. The authors obtain a new theoretical approximation formula for the upper bound of the total error of the numerical solution, as well as theoretical formulae for the optimal grid size and time step. The reliability of the obtained formulae is demonstrated with numerical experimental examples. Next, the ratio of the optimal time steps under two different machine precisions is found to satisfy a universal relation that depends only on the machine precision involved. Finally, theoretical verification suggests that this problem satisfies the computational uncertainty principle when the grid ratio is fixed, demonstrating the inevitable existence of an optimal time step size under a finite machine precision.摘要本文对于应用Lax- Friedrichs 格式数值求解一维平流方程, 研究数值求解过程中产生的截断误差与舍入误差, 以及两种误差逐层向高时间层传播的累积, 得到新的数值解总误差上界的理论近似公式, 以及最优格距和最优时间步长的理论公式. 通过数值算例验证了所得公式的可靠性. 然后, 发现了两种不同机器精度下最优时间步长之比满足的一个仅与机器精度有关的普适关系. 最后, 理论验证了在网格比固定的情况下, 此问题满足数值计算的不确定性原理, 以及在机器有限精度下最优时间步长的必然存在.

Avaliação de formulações de taxa de deformação através de escoamentos permanentes e uniformes de fluidos não-Newtonianos no contexto de equações de águas rasas

ABSTRACT Non-Newtonian rheology effects, such as pseudoplasticity and viscoplasticity, are understood as shear stresses, incorporated to the energy slope term in the Shallow-Water Equations (SWE). However, non-Newtonian shear stresses are dependent of the shear rate, whose formulation is a function of the gradient of the velocity profile in the bottom. This study investigated two shear rate formulations that are commonly applied in the SWE literature: 1) a non-parameterized function; and 2) a function based on the Herschel-Bulkley rheological model. Their influence in steady uniform flows of non-Newtonian fluids was evaluated through numerical-theoretical comparisons. A Lax-Friedrichs scheme was implemented to solve the SWE system and allowed employing the shear rate formulations. Experimental tests were carried out and numerical simulations of hypothetical scenarios were performed. It was found that the non-parameterized formulation presented deviation in normal depth up to 14% in comparison with theoretical solution, while the formulation based on the Herschel-Bulkley model provided a good agreement, corroborated by punctual Computational Fluid Dynamics simulations (deviation less than 2%) and experimental data. The ratio of both shear rate formulations is strongly correlated to the deviation of normal depth, indicating that the non-parameterized shear rate function does not provide an acceptable result in the steady uniform flow.

SPH modeling of substance transport in flows with large deformation

The velocity field in coastal and oceanic currents is mostly non-uniform, which will result in irregular particle distribution when the fluid is represented by an amount of moving discrete particles as in smoothed particle hydrodynamics (SPH). When the non-uniformity of the flow is big, i.e., with large deformation, the conventional SPH method can hardly solve the associated advection-diffusion process (e.g., substance transport). To accurately simulate the substance transport in flows with large deformation, two types of particle shifting techniques (PSTs) are incorporated into the conventional SPH in this paper. One is based on current particle distance, and the other is based on Fick’s law. In the second type, the repulsive force (RF) term for suppressing the paring instability that occurs in particle shifting technique (PST) is studied and the effect of the kernel function is examined. By introducing a particle disorder measurement, the simulated results of SPH with the two types of PSTs and their modifications are evaluated and the influence of the shifting magnitude is analyzed. The suggestions for how to set reasonable parameters in PSTs are provided by a systematic parametric study. For further illustration, the simulation of the anisotropic diffusion is also examined. To give reliable reference solutions, the high-resolution modified total variation diminishing Lax Friedrichs scheme with Superbee limiter (MTVDLF-Superbee) with fine mesh is also implemented. The validated Lagrangian particle model with optimized PST is applied to a practical application.

An efficient invariant-region-preserving central scheme for hyperbolic conservation laws

Due to the Riemann solver free and avoiding characteristic decomposition, the central scheme is a simple and efficient tool for numerical solution of hyperbolic conservation laws (Nessyahu and Tadmor, J. Comput. Phys., 87(2):314-329,1990). But the theoretical Courant number CFL in order to preserve the invariant region of the numerical solution is very small, and there is lack of the stability proof for nonlinear systems. By adding a limiter on the reconstructed slope without requiring clipping condition, we enlarge the value of the CFL to admit larger time step. Then a widely applicable stability proof, which is suitable for general hyperbolic conservation laws, is given by writing the evolved solution as convex combinations in terms of the Lax-Friedrichs scheme. Some numerical experiments are carried out to verify the robustness.

Effects of Air Quality on Housing Location: A Predictive Dynamic Continuum User-Optimal Approach

Recent decades have seen increasing concerns regarding air quality in housing locations. This study proposes a predictive continuum dynamic user-optimal model with combined choice of housing location, destination, route, and departure time. A traveler’s choice of housing location is modeled by a logit-type demand distribution function based on air quality, housing rent, and perceived travel costs. Air quality, or air pollutants, within the modeling region are governed by the vehicle-emission model and the advection-diffusion equation for dispersion. In this study, the housing-location problem is formulated as a fixed-point problem and the predictive continuum dynamic user-optimal model with departure-time consideration is formulated as a variational inequality problem. The Lax-Friedrichs scheme, the fast-sweeping method, the Goldstein-Levitin-Polyak projection algorithm, and self-adaptive successive averages are adopted to discretize and solve these problems. A numerical example is given to demonstrate the characteristics of the proposed housing-location choice problem with consideration of air quality and to demonstrate the effectiveness of the solution algorithms.

A class of high-order weighted compact central schemes for solving hyperbolic conservation laws

We propose a class of weighted compact central schemes for solving hyperbolic conservation laws. The linear version can be considered as a high-order extension of the central Lax–Friedrichs scheme and the central conservation element and solution element scheme. On every cell, the solution is approximated by a Pth-order polynomial of which all the DOFs are stored and updated separately. The cell average is updated by a classical finite volume scheme which is constructed based on space-time staggered meshes such that the fluxes are continuous across the interfaces of the adjacent control volumes and, therefore, the local Riemann problem is bypassed. The kth-order spatial derivatives are updated by a central difference of the (k−1)th-order spatial derivatives at cell vertices. All the space-time information is calculated by the Cauchy–Kovalewski procedure. By doing so, the schemes are able to achieve arbitrarily uniform space-time high-order on a compact stencil consisting of only neighboring cells with only one explicit time step. In order to capture discontinuities without spurious oscillations, a weighted essentially non-oscillatory type limiter is tailor-made for the schemes. The limiter preserves the compactness and high-order accuracy of the schemes. The schemes' accuracy, robustness, and efficiency are verified by several numerical examples of scalar conservation laws and the compressible Euler equations.

Weak solutions to the Cauchy problem of the time-dependent Thomas–Fermi equations

In this paper, we are concerned with the existence of weak solutions of the time-dependent Thomas–Fermi equations. We derive approximate solutions by the fractional step Lax–Friedrichs scheme and establish uniform boundedness of approximate solutions. Based on the uniform energy-type estimates, we establish that the entropy dissipation measures of the weak solution of the one-dimensional time-dependent Thomas–Fermi equations for weak entropy–entropy flux pairs, generated by compactly supported C0∞ test functions, are confined in a compact set in Hloc−1. We prove that the Young measure must be a Dirac measure by the Tartar–Murat commutator relation. The convergence of approximate solutions is established by using the compensated compactness method.

A Posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

The paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for $n\times n$ hyperbolic conservation laws in one space dimension. These estimates are achieved by a "post-processing algorithm", checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained by the Godunov or the Lax-Friedrichs scheme, backward Euler approximations, and the method of periodic smoothing. Some numerical implementations are presented.

Nonlinear Dirac Equation for Graphene

The possibility of introducing a nonlinear correction to the Dirac equation for graphene in order to adequately describe collective electronic phenomena is considered. In contrast to the other papers on this topic the interaction term includes the sum of the spinor components’ squares instead of their difference. Particular attention is paid to the equality of the spatial coordinates. We investigate the properties of the obtained nonlinear equation, in order to describe the high-temperature ferromagnetism in graphene without making any assumptions on the key role of defects of the structure in providing the effect. The numerical simulation is carried out for the simple boundary and initial conditions with the Lax–Friedrichs scheme as a result of which information is obtained on the dynamics of the electron density for a number of the simplest initial and boundary conditions.

Flood risk mapping using uncertainty propagation analysis on a peak discharge: case study of the Mille Iles River in Quebec

This study uses uncertainty propagation in real flood events to derive a probabilistic flood map. The flood event of spring 2017 in Quebec was selected for this analysis, with the computational domain being a reach of the Mille Iles River. The main parameter deemed uncertain in this work is the upstream water discharge; a given value of this discharge is utilized to build a random sample of 500 scenarios using the Latin hypercube sampling method. Simulations were run using CuteFlow-Cuda, an in-house finite volume-based shallow water equations solver, to derive the statistical mean and the standard deviation of the free surface elevation and the water depth at each node. For this real flood case, the initial interface flux scheme had to be adapted, combining a developed version of the scheme introduced by Harten, Lax and van Leer at wet interfaces and the Lax–Friedrichs scheme with additional free surface corrections for wet and dry transitions. Comparisons with results obtained from TELEMAC and from in situ observations show generally close predictions, and overall good agreement with observations. Errors of the free surface prediction relative to observations are less than 2.75%. A map based on the standard deviation of the water depth is presented to enhance the areas most prone to flooding. Finally, a flood map is produced, showing the flooded inhabited areas near the municipalities of Saint-Eustache and Deux Montagnes around the reach of the Mille Iles River as it overflows its natural bed.

Remarks on the energy inequality of a global $L^\infty$ solution to the compressible Euler equations for the isentropic nozzle flow

We study the compressible Euler equations in the isentropic nozzle flow. The global existence of an $L^{\infty}$ solution has been proved in (Tsuge in Nonlinear Anal. Real World Appl. 209: 217-238 (2017)) for large data and general nozzle. However, unfortunately, this solution does not possess finiteness of energy. Although the modified Godunov scheme is introduced in this paper, we cannot deduce the energy inequality for the approximate solutions. Therefore, our aim in the present paper is to derive the energy inequality for an $L^{\infty}$ solution. To do this, we introduce the modified Lax Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. We shall first deduce from the formula the energy inequality. Next, applying the compensated compactness method, the approximate solution converges to a weak solution. The energy inequality also holds for the solution as the limit. As a result, since our solutions are $L^{\infty}$, they possess finite energy and propagation, which are essential to physics.

Modelling and simulation of a wave energy converter

In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions.

A survey of numerical schemes for transportation equation

The convection-diffusion equation is a fundamental equation that exists widely. The convection-diffusion equation consists of two processes: diffusion and convection. The convection-diffusion equation can also be called drift-diffusion equaintion. The convection – diffusion equation mainly characterizes natural phenomenon in which physical particles, energy are transferred in a system. The well-known linear transport equation is also one kind of convection-diffusion equation. The transport equation can describe the transport of a scalar field such as material feature, chemical reaction or temperature in an incompressible flow. In this paper, we discuss the famous numerical scheme, Lax-Friedrichs method, for the linear transport equation. The important ingredient of the design of the Lax-Friedrichs Method, namely the choice of the numerical fluxes will be discussed in detail. We give a detailed proof of the L1 stability of the Lax-Friedrichs scheme for the linear transport equation. We also address issues related to the implementation of this numerical scheme.

Studying perturbations and wave propagations by lane closures on traffic characteristics based on a dynamic approach

Lane closures cause perturbations and wave propagations on traffic characteristics of freeways and consequently affect traffic flow. To conduct a dynamic analysis regarding the effects of lane closures on traffic characteristics, this study aimed to examine the effects of lane closures as multi-static bottlenecks on the evolution and wave propagation of uniform traffic flow in a four-lane divided freeway based on the field study and macroscopic traffic flow model. One-lane closure is a work zone and two-lane closure occurs when the second lane near the one-lane closure is blocked due to an incident caused by a lane change. In addition, the classical Lighthill–Whitham–Richards (LWR) model was combined with the non-linear Greenshield’s speed–density relation to expand the models for characterizing the behavior of shockwaves under lane closures. Thus, based on numerical procedures regarding the Lax–Friedrichs scheme, the results indicated that wave propagations by two-lane closure have significant effects on the perturbations of traffic flow density which are 1.5 times as much as the wave propagations in one-lane closure. Moreover, one-lane closure reduced the capacity by 21% although the two-lane closure caused a 45% decrease in the capacity. A comparison between expanded LWRρ and LWRu models with the LWR model demonstrated that expanded models under two-lane closure and one-lane closure improved LWR model for the congested condition, and between uncongested and congested traffic condition 15.61% and 39%, respectively. Therefore, expanded models revealed a better performance as a dynamic approach to analyze the behavior of wave propagations in jam traffic conditions compared to LWR model under lane closures.

A two-dimensional multi-class traffic flow model

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [ 21 ], where a two-dimensional model for a single class of vehicles is proposed, we extend the dynamics to a multi-class model leading to a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems we also present a Lax-Friedrichs type discretization scheme recovering the theoretical results by means of numerical tests. We calibrate the multi-class model with real data and compare the fitted model to the real trajectories. Finally, we test the ability of the model to simulate the overtaking of vehicles.

Stable Algorithm Based On Lax-Friedrichs Scheme for Visual Simulation of Shallow Water

Many game applications require fluid flow visualization of shallow water, especially dam-break flow. A Shallow Water Equation (SWE) is a mathematical model of shallow water flow which can be used to compute the flow depth and velocity. We propose a stable algorithm for visualization of dam-break flow on flat and flat with bumps topography. We choose Lax-Friedrichs scheme as the numerical method for solving the SWE. Then, we investigate the consistency, stability, and convergence of the scheme. Finally, we transform the strategy into a visualization algorithm of SWE and analyze the complexity. The results of this paper are: 1) the Lax-Friedrichs scheme that is consistent and conditionally stable; furthermore, if the stability condition is satisfied, the scheme is convergent; 2) an algorithm to visualize flow depth and velocity which has complexity O(N) in each time iteration. We have applied the algorithm to flat and flat with bumps topography. According to visualization results, the numerical solution is very close to analytical solution in the case of flat topography. In the case of flat with bumps topography, the algorithm can visualize the dam-break flow and after a long time the numerical solution is very close to the analytical steady-state solution. Hence the proposed visualization algorithm is suitable for game applications containing flat with bumps environments.