Articles de revues : « Lax-Friedrichs method »

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Shampine, L. F. "Two-step Lax–Friedrichs method." Applied Mathematics Letters 18, no. 10 (2005): 1134–36. http://dx.doi.org/10.1016/j.aml.2004.11.007.

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Yu, Simin. "A survey of numerical schemes for transportation equation." E3S Web of Conferences 308 (2021): 01020. http://dx.doi.org/10.1051/e3sconf/202130801020.

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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.

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Breuß, Michael. "The correct use of the Lax–Friedrichs method." ESAIM: Mathematical Modelling and Numerical Analysis 38, no. 3 (2004): 519–40. http://dx.doi.org/10.1051/m2an:2004027.

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Sharma, Deepika, and Kavita Goyal. "Wavelet optimized upwind conservative method for traffic flow problems." International Journal of Modern Physics C 31, no. 06 (2020): 2050086. http://dx.doi.org/10.1142/s0129183120500862.

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Numerical schemes, namely, upwind nonconservative, upwind conservative, Lax–Friedrichs, Lax–Wendroff, MacCormack and Godunov are applied and compared on traffic flow problems. The best scheme, namely, upwind conservative is used for wavelet-optimized method using Daubechies wavelet for numerically solving the same traffic flow problems. Numerical results corresponding to the traffic flow problem with the help of wavelet-optimized, adaptive grid, upwind conservative method have been given. Moreover, the run time carried out by the developed technique have been compared to that of run time carried out by finite difference technique. It is observed that, in terms of run time, the proposed method performs better.

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Chatterjee, N., and U. S. Fjordholm. "A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws." IMA Journal of Numerical Analysis 40, no. 1 (2018): 405–21. http://dx.doi.org/10.1093/imanum/dry074.

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Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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Araujo, Isamara L. N., Panters Rodríguez-Bermúdez, and Yoisell Rodríguez-Núñez. "Numerical Study for Two-Phase Flow with Gravity in Homogeneous and Piecewise-Homogeneous Porous Media." TEMA (São Carlos) 21, no. 1 (2020): 21. http://dx.doi.org/10.5540/tema.2020.021.01.21.

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In this work we study two-phase flow with gravity either in 1-rock homogeneous media or 2-rocks composed media, this phenomenon can be modeled by a non-linear scalar conservation law with continuous flux function or discontinuous flux function, respectively. Our study is essentially from a numerical point of view, we apply the new Lagrangian-Eulerian finite difference method developed by Abreu and Pérez and the Lax-Friedrichs classic method to obtain numerical entropic solutions. Comparisons between numerical and analytical solutions show the efficiency of the methods even for discontinuous flux function.

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Wang, Min, and Xiaohua Zhang. "A High–Order WENO Scheme Based on Different Numerical Fluxes for the Savage–Hutter Equations." Mathematics 10, no. 9 (2022): 1482. http://dx.doi.org/10.3390/math10091482.

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The study of rapid free surface granular avalanche flows has attracted much attention in recent years, which is widely modeled using the Savage–Hutter equations. The model is closely related to shallow water equations. We employ a high-order shock-capturing numerical model based on the weighted essential non-oscillatory (WENO) reconstruction method for solving Savage–Hutter equations. Three numerical fluxes, i.e., Lax–Friedrichs (LF), Harten–Lax–van Leer (HLL), and HLL contact (HLLC) numerical fluxes, are considered with the WENO finite volume method and TVD Runge–Kutta time discretization for the Savage–Hutter equations. Numerical examples in 1D and 2D space are presented to compare the resolution of shock waves and free surface capture. The numerical results show that the method proposed provides excellent performance with high accuracy and robustness.

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Ali, Ali Hasan, Ahmed Shawki Jaber, Mustafa T. Yaseen, Mohammed Rasheed, Omer Bazighifan, and Taher A. Nofal. "A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model." Complexity 2022 (June 27, 2022): 1–9. http://dx.doi.org/10.1155/2022/9367638.

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In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.

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Setiyowati, R., and Sumardi. "A Simulation of Shallow Water Wave Equation Using Finite Volume Method: Lax-Friedrichs Scheme." Journal of Physics: Conference Series 1306 (August 2019): 012022. http://dx.doi.org/10.1088/1742-6596/1306/1/012022.

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Kao, Chiu Yen, Carmeliza Navasca, and Stanley Osher. "The Lax–Friedrichs sweeping method for optimal control problems in continuous and hybrid dynamics." Nonlinear Analysis: Theory, Methods & Applications 63, no. 5-7 (2005): e1561-e1572. http://dx.doi.org/10.1016/j.na.2005.01.061.

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Arry Sanjoyo, Bandung, Mochamad Hariadi, and Mauridhi Hery Purnomo. "Stable Algorithm Based On Lax-Friedrichs Scheme for Visual Simulation of Shallow Water." EMITTER International Journal of Engineering Technology 8, no. 1 (2020): 19–34. http://dx.doi.org/10.24003/emitter.v8i1.479.

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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.

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Krasnov, Mikhail Mikhailovich, Marina Eugenievna Ladonkina, Olga Alexandrovna Neklyudova, and Vladimir Fedorovich Tishkin. "On the influence of the choice of the numerical flow on the solution of problems with shock waves by the discontinuous Galerkin method." Keldysh Institute Preprints, no. 91 (2022): 1–21. http://dx.doi.org/10.20948/prepr-2022-91.

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The paper compares various numerical flows in the calculation of flows with the presence of shock waves by first-order schemes and second-order DG method on schemes from the Quirk’s list, namely: the Quirk problem and its modifications, shock wave diffraction at a 90 degree angle, the problem of double Mach reflection. It is shown that the use of HLLC and Godunov’s numerical flows in calculations can lead to instability, the Rusanov-Lax-Friedrichs flow leads to high dissipation of the calculation. The most versatile in carrying out production calculations are hybrid numerical flows, which allow suppressing the development of instabilities and maintaining the accuracy of the method.

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Xu, J., M. Luo, Z. Wu, S. Wang, B. Qi, and Z. Qiao. "Pressure and Temperature Prediction of Transient Flow in HTHP Injection Wells by Lax-Friedrichs Method." Petroleum Science and Technology 31, no. 9 (2013): 960–76. http://dx.doi.org/10.1080/10916466.2010.535083.

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Rossi, Elena, Jennifer Weißen, Paola Goatin, and Simone Göttlich. "Well-posedness of a non-local model for material flow on conveyor belts." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 2 (2020): 679–704. http://dx.doi.org/10.1051/m2an/2019062.

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In this paper, we focus on finite volume approximation schemes to solve a non-local material flow model in two space dimensions. Based on the numerical discretisation with dimensional splitting, we prove the convergence of the approximate solutions, where the main difficulty arises in the treatment of the discontinuity occurring in the flux function. In particular, we compare a Roe-type scheme to the well-established Lax–Friedrichs method and provide a numerical study highlighting the benefits of the Roe discretisation. Besides, we also prove the L1-Lipschitz continuous dependence on the initial datum, ensuring the uniqueness of the solution.

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Frenzel, David, and Jens Lang. "A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws." Computational Optimization and Applications 80, no. 1 (2021): 301–20. http://dx.doi.org/10.1007/s10589-021-00295-2.

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AbstractThe weighted essentially non-oscillatory (WENO) methods are popular and effective spatial discretization methods for nonlinear hyperbolic partial differential equations. Although these methods are formally first-order accurate when a shock is present, they still have uniform high-order accuracy right up to the shock location. In this paper, we propose a novel third-order numerical method for solving optimal control problems subject to scalar nonlinear hyperbolic conservation laws. It is based on the first-disretize-then-optimize approach and combines a discrete adjoint WENO scheme of third order with the classical strong stability preserving three-stage third-order Runge–Kutta method SSPRK3. We analyze its approximation properties and apply it to optimal control problems of tracking-type with non-smooth target states. Comparisons to common first-order methods such as the Lax–Friedrichs and Engquist–Osher method show its great potential to achieve a higher accuracy along with good resolution around discontinuities.

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Vosoughifar, Hamid Reza, Azam Dolatshah, and Seyed Kazem Sadat Shokouhi. "Discretization of Multidimensional Mathematical Equations of Dam Break Phenomena Using a Novel Approach of Finite Volume Method." Journal of Applied Mathematics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/642485.

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This paper was concerned to simulate both wet and dry bed dam break problems. A high-resolution finite volume method (FVM) was employed to solve the one-dimensional (1D) and two-dimensional (2D) shallow water equations (SWEs) using an unstructured Voronoi mesh grid. In this attempt, the robust local Lax-Friedrichs (LLxF) scheme was used for the calculating of the numerical flux at cells interfaces. The model named V-Break was run under the asymmetry partial and circular dam break conditions and then verified by comparing the model outputs with the documented results. Due to a precise agreement between those output and documented results, the V-Break could be considered as a reliable method for dealing with shallow water (SW) and shock problems, especially those having discontinuities. In addition, statistical observations indicated a good conformity between the V-Break and analytical results clearly.

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Wang, Shu, and Yabo Ren. "Weak solutions to the Cauchy problem of the time-dependent Thomas–Fermi equations." Journal of Mathematical Physics 63, no. 6 (2022): 061507. http://dx.doi.org/10.1063/5.0082846.

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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 [Formula: see text] test functions, are confined in a compact set in [Formula: see text]. 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.

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Outada, Nisrine, Nicolas Vauchelet, Thami Akrid, and Mohamed Khaladi. "From kinetic theory of multicellular systems to hyperbolic tissue equations: Asymptotic limits and computing." Mathematical Models and Methods in Applied Sciences 26, no. 14 (2016): 2709–34. http://dx.doi.org/10.1142/s0218202516500640.

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This paper deals with the analysis of the asymptotic limit toward the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory. After having chosen an appropriate scaling of time and space, a Chapman–Enskog expansion is combined with a closed, by minimization, technique to derive hyperbolic models at the macroscopic level. The resulting macroscopic equations show how the macroscopic tissue behavior can be described by hyperbolic systems which seem the most natural in this context. We propose also an asymptotic-preserving well-balanced scheme for the one-dimensional hyperbolic model, in the two-dimensional case, we consider a time-splitting method between the conservative part and the source term where the conservative equation is approximated by the Lax–Friedrichs scheme.

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Zhou, Shengcheng, Zhipeng Li, and Li Deng. "Spatial convergence study of Lax-Friedrichs WENO fast sweeping method on the SN transport equation with nonsmoothness." Annals of Nuclear Energy 166 (February 2022): 108707. http://dx.doi.org/10.1016/j.anucene.2021.108707.

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Mohamed, Kamel, Hanan A. Alkhidhr, and Mahmoud A. E. Abdelrahman. "The NHRS scheme for the Chaplygin gas model in one and two dimensions." AIMS Mathematics 7, no. 10 (2022): 17785–801. http://dx.doi.org/10.3934/math.2022979.

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<abstract>The main motive of this work is to introduce a numerical investigation for the one and two-dimensional (1D/2D) Chaplygin gas model. Namely, we developed the non homogeneous Riemann solver (NHRS) method to solve these models. After discussing the Chaplygin gas models and the numerical scheme, various 1D and 2D test problems are introduced. In order to complete the numerical investigation in a completely unified way, Rusanov scheme, modified Lax-Friedrichs and analytical solution are compared with NHRS scheme in 1D case. The acquired results clarify the high resolution of the NHRS technique. The NHRS technique is efficacious and robust. Finally, our study displays that the NHRS scheme is a very powerful tool to solve many other models arising in applied science.</abstract>

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Hong, Chengyu, Xuben Wang, Gaishan Zhao, et al. "Discontinuous finite element method for efficient three-dimensional elastic wave simulation." Journal of Geophysics and Engineering 18, no. 1 (2021): 98–112. http://dx.doi.org/10.1093/jge/gxaa070.

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Abstract The existing discontinuous Galerkin (DG) finite element method (FEM) for the numerical simulation of elastic wave propagation is primarily implemented in two dimensions. Here, a discontinuous FEM (DFEM) for efficient three-dimensional (3D) elastic wave simulation is presented. First, the velocity–stress equations of 3D elastic waves in isotropic media are transformed into first-order coefficient-changed partial differential equations. A DG discretisation method for wave field values on a unit boundary is then defined using the local Lax–Friedrichs flux format. The equations are first transformed into equivalent integral equations, and subsequently into a spatial semi-discrete ordinary differential equation system using a hierarchical orthogonal basis function. The DFEM is extended to an arbitrary high-order accuracy in the time domain using the exponential integrator technique and the explicit optimal strong-stability-preserving Runge–Kutta method. Finally, an efficient method for selecting the calculation area of the geometry of the current shot record is realised. For the computation, a multi-node parallelism with improved resource utilisation and parallelisation efficiency is implemented. The numerical results show that the proposed method can improve both the accuracy of the simulation and the efficiency of the calculation compared with existing methods.

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Xu, Jiuping, Min Luo, Jiancheng Hu, Shize Wang, Bin Qi, and Zhiguo Qiao. "A Direct Eulerian GRP Scheme for the Prediction of Gas-Liquid Two-Phase Flow in HTHP Transient Wells." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/171732.

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A coupled system model of partial differential equations is presented in this paper, which concerns the variation of the pressure and temperature, velocity, and density at different times and depths in high temperature-high pressure (HTHP) gas-liquid two-phase flow wells. A new dimensional splitting technique with Eulerian generalized riemann problem (GRP) scheme is applied to solve this set of conservation equations, where Riemann invariants are introduced as the main ingredient to resolve the generalized Riemann problem. The basic data of “X well” (HTHP well), 7100 m deep, located in Southwest China, is used for the case history calculations. Curve graphs of pressures and temperatures along the depth of the well are plotted at different times. The comparison with the results of Lax Friedrichs (LxF) method shows that the calculating results are more fitting to the values of real measurement and the new method is of high accuracy.

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Martínez-Aranda, S., A. Ramos-Pérez, and P. García-Navarro. "A 1D shallow-flow model for two-layer flows based on FORCE scheme with wet–dry treatment." Journal of Hydroinformatics 22, no. 5 (2020): 1015–37. http://dx.doi.org/10.2166/hydro.2020.002.

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Abstract The two-layer problem is defined as the coexistence of two immiscible fluids, separated by an interface surface. Under the shallow-flow hypothesis, 1D models are based on a four equations system accounting for the mass and momentum conservation in each fluid layer. Mathematically, the system of conservation laws modelling 1D two-layer flows has the important drawback of loss of hyperbolicity, causing that numerical schemes based on the eigenvalues of the Jacobian become unstable. In this work, well-balanced FORCE scheme is proposed for 1D two-layer shallow flows. The FORCE scheme combines the first-order Lax–Friedrichs flux and the second-order Lax–Wendroff flux. The scheme is supplemented with a hydrostatic reconstruction procedure in order to ensure the well-balanced behaviour of the model for steady flows even under wet–dry conditions. Additionally, a method to obtain high-accuracy numerical solutions for two-layer steady flows including friction dissipation is proposed to design reference benchmark tests for model validation. The enhanced FORCE scheme is faced to lake-at-rest benchmarking tests and steady flow cases including friction, demonstrating its well-balanced character. Furthermore, the numerical results obtained for highly unsteady two-layer dambreaks are used to analyse the robustness and accuracy of the model under a wide range of flow conditions.

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Amundsen, Lasse, and Ørjan Pedersen. "Time step n-tupling for wave equations." GEOPHYSICS 82, no. 6 (2017): T249—T254. http://dx.doi.org/10.1190/geo2017-0377.1.

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We have constructed novel temporal discretizations for wave equations. We first select an explicit time integrator that is of second order, leading to classic time marching schemes in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. Then, we determine how the time step can be doubled, tripled, or generally, [Formula: see text]-tupled, producing a new time-stepping method in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. In-between time values of the wavefield are eliminated. Using the Fourier method to calculate space derivatives, the new time integrators allow larger stable time steps than traditional time integrators; however, like the Lax-Wendroff procedure, they require more computational effort per time step. Because the new schemes are developed from the classic second-order time-stepping scheme, they will have the same properties, except the Courant-Friedrichs-Lewy stability condition, which becomes relaxed by the factor [Formula: see text] compared with the classic scheme. As an example, we determine the method for solving scalar wave propagation in which doubling the time step is 15% faster than a Lax-Wendroff correction scheme of the same spatial order because it can increase the time step by [Formula: see text] only.

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Palm, Johannes, and Claes Eskilsson. "Influence of Bending Stiffness on Snap Loads in Marine Cables: A Study Using a High-Order Discontinuous Galerkin Method." Journal of Marine Science and Engineering 8, no. 10 (2020): 795. http://dx.doi.org/10.3390/jmse8100795.

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Marine cables are primarily designed to support axial loads. The effect of bending stiffness on the cable response is therefore often neglected in numerical analysis. However, in low-tension applications such as umbilical modelling of ROVs or during slack events, the bending forces may affect the slack regime dynamics of the cable. In this paper, we present the implementation of bending stiffness as a rotation-free, nested local Discontinuous Galerkin (DG) method into an existing Lax–Friedrichs-type solver for cable dynamics based on an hp-adaptive DG method. Numerical verification shows exponential convergence of order P and P+1 for odd and even polynomial orders, respectively. Validation of a swinging cable shows good comparison with experimental data, and the importance of bending stiffness is demonstrated. Snap load events in a deep water tether are compared with field-test data. The bending forces affect the low-tension response for shorter lengths of tether (200–500 m), which results in an increasing snap load magnitude for increasing bending stiffness. It is shown that the nested LDG method works well for computing bending effects in marine cables.

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Hu, Jiangtao, Jianliang Qian, Jian Song, Min Ouyang, Junxing Cao, and Shingyu Leung. "Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media." GEOPHYSICS 86, no. 4 (2021): T179—T192. http://dx.doi.org/10.1190/geo2020-0659.1.

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Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we have developed a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classic real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially nonoscillatory schemes. Numerical examples demonstrate that our method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. Our methods can be useful for migration and tomography in attenuating media.

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Scandaliato, Angelo L., and Meng-Sing Liou. "AUSM-Based High-Order Solution for Euler Equations." Communications in Computational Physics 12, no. 4 (2012): 1096–120. http://dx.doi.org/10.4208/cicp.250311.081211a.

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AbstractIn this paper we demonstrate the accuracy and robustness of combining the advection upwind splitting method (AUSM), specifically AUSM+-UP, with high-order upwind-biased interpolation procedures, the weighted essentially non-oscillatory (WENO-JS) scheme and its variations, and the monotonicity preserving (MP) scheme, for solving the Euler equations. MP is found to be more effective than the three WENO variations studied. AUSM+-UP is also shown to be free of the so-called “carbuncle” phenomenon with the high-order interpolation. The characteristic variables are preferred for interpolation after comparing the results using primitive and conservative variables, even though they require additional matrix-vector operations. Results using the Roe flux with an entropy fix and the Lax-Friedrichs approximate Riemann solvers are also included for comparison. In addition, four reflective boundary condition implementations are compared for their effects on residual convergence and solution accuracy. Finally, a measure for quantifying the efficiency of obtaining high order solutions is proposed; the measure reveals that a maximum return is reached after which no improvement in accuracy is possible for a given grid size.

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Keshari, Ashok K., Deba P. Satapathy, and Amod Kumar. "The influence of vertical density and velocity distributions on snow avalanche runout." Annals of Glaciology 51, no. 54 (2010): 200–206. http://dx.doi.org/10.3189/172756410791386409.

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AbstractA one-dimensional avalanche dynamics model accounting for vertical density and velocity distributions is presented. Mass and momentum flux distribution factors are derived to incorporate the effect of density and velocity variations within the framework of depth-integrated models. Using experiments of avalanche flows on an inclined snow chute at Dhundhi, Manali, India, we conceptualize snow flow rheology as a Voellmy fluid where the distribution of internal shearing is given by a Newtonian fluid (NF) or Criminale–Ericksen–Filbey fluid (CEFF). Then the generalized mass and momentum distribution factors are computed for these two fluid models for different density stratifications. Numerical solutions are obtained using a total variation diminishing Lax–Friedrichs (TVDLF) finite-difference method. The model is validated with the experimental results. We find that the flow features of the chute experiments are simulated well by the model. The velocities and runout distances are obtained for the Voellmy model with both NF and CEFF extensions for various input volumes, and the optimum values of the model parameters, namely, coefficients of dynamic and turbulent friction, are determined.

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Mo, Tiexiang, and Guodong Li. "Parallel Accelerated Fifth-Order WENO Scheme-Based Pipeline Transient Flow Solution Model." Applied Sciences 12, no. 14 (2022): 7350. http://dx.doi.org/10.3390/app12147350.

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The water hammer phenomenon is the main problem in long-distance pipeline networks. The MOC (Method of characteristics) and finite difference methods lead to severe constraints on the mesh and Courant number, while the finite volume method of the second-order Godunov scheme has limited intermittent capture capability. These methods will produce severe numerical dissipation, affecting the computational efficiency at low Courant numbers. Based on the lax-Friedrichs flux splitting method, combined with the upstream and downstream virtual grid boundary conditions, this paper uses the high-precision fifth-order WENO scheme to reconstruct the interface flux and establishes a finite volume numerical model for solving the transient flow in the pipeline. The model adopts the GPU parallel acceleration technology to improve the program’s computational efficiency. The results show that the model maintains the excellent performance of intermittent excitation capture without spurious oscillations even at a low Courant number. Simultaneously, the model has a high degree of flexibility in meshing due to the high insensitivity to the Courant number. The number of grids in the model can be significantly reduced and higher computational efficiency can be obtained compared with MOC and the second-order Godunov scheme. Furthermore, this paper analyzes the acceleration effect in different grids. Accordingly, the acceleration effect of the GPU technique increases significantly with the increase in the number of computational grids. This model can support efficient and accurate fast simulation and prediction of non-constant transient processes in long-distance water pipeline systems.

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Ariunaa, U., M. Dumbser, and Ts Sarantuya. "Complete Riemann Solvers for the Hyperbolic GPR Model of Continuum Mechanics." Bulletin of Irkutsk State University. Series Mathematics 35 (2021): 60–72. http://dx.doi.org/10.26516/1997-7670.2021.35.60.

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In this paper, complete Riemann solver of Osher-Solomon and the HLLEM Riemann solver for unified first order hyperbolic formulation of continuum mechanics, which describes both of fluid and solid dynamics, are presented. This is the first time that these types of Riemann solvers are applied to such a complex system of governing equations as the GPR model of continuum mechanics. The first order hyperbolic formulation of continuum mechanics recently proposed by Godunov S. K., Peshkov I. M. and Romenski E. I., further denoted as GPR model includes a hyperbolic formulation of heat conduction and an overdetermined system of PDE. Path-conservative schemes are essential in order to give a sense to the non-conservative terms in the weak solution framework since governing PDE system contains non-conservative products, too. New Riemann solvers are implemented and tested successfully, which means it certainly acts better than standard local Lax-Friedrichs-type or Rusanov-type Riemann solvers. Two simple computational examples are presented, but the obtained computational results clearly show that the complete Riemann solvers are less dissipative than the simple Rusanov method that was employed in previous work on the GPR model.

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Bürger, Raimund, Harold Deivi Contreras, and Luis Miguel Villada. "A Hilliges-Weidlich-type scheme for a one-dimensional scalar conservation law with nonlocal flux." Networks and Heterogeneous Media 18, no. 2 (2023): 664–93. http://dx.doi.org/10.3934/nhm.2023029.

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<abstract>The simulation model proposed in [M. Hilliges and W. Weidlich. A phenomenological model for dynamic traffic flow in networks. <italic>Transportation Research Part B: Methodological</italic>, <bold>29</bold> (6): 407–431, 1995] can be understood as a simple method for approximating solutions of scalar conservation laws whose flux is of density times velocity type, where the density and velocity factors are evaluated on neighboring cells. The resulting scheme is monotone and converges to the unique entropy solution of the underlying problem. The same idea is applied to devise a numerical scheme for a class of one-dimensional scalar conservation laws with nonlocal flux and initial and boundary conditions. Uniqueness of entropy solutions to the nonlocal model follows from the Lipschitz continuous dependence of a solution on initial and boundary data. By various uniform estimates, namely a maximum principle and bounded variation estimates, along with a discrete entropy inequality, the sequence of approximate solutions is shown to converge to an entropy weak solution of the nonlocal problem. The improved accuracy of the proposed scheme in comparison to schemes based on the Lax-Friedrichs flux is illustrated by numerical examples. A second-order scheme based on MUSCL methods is presented.</abstract>

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Hanh, Nguyen Van, Nguyen Van Diep, and Ngo Huy Can. "On some numerical methods for solving the 1-D Saint-Venant equations of general flow regime. Part 1: Numerical methods." Vietnam Journal of Mechanics 24, no. 4 (2002): 236–48. http://dx.doi.org/10.15625/0866-7136/24/4/6623.

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Development of methods for numerical simulation of dike- or dam-break flood is one of essential problems of Fluid Mechanics at the present time. Many numerical methods for solving the 1-D Saint-Venant equations have been proposed. However, the analysis, the evaluation and the selection of appropriate and efficient methods are interest of many research groups and institutions in the world. The purpose of this paper is to introduce and to evaluate four numerical methods for solving the 1-D homogenous Saint-Venant equations in combination with three approaches of processing source terms. The evaluation is based on the test problems, proposed by European Hydraulic Research Laboratories. The Part 1of the paper presents some modern numerical methods for solving the 1-D Saint-Venant equations of general Bow regime, where the Bow may be mixed between sub critical and supercritical. The homogenous part of the system of equations is numerically solved by "shock capturing methods" for conservation laws: the Lax-Friedrichs, the self adjusting hybrid, the Roe's approximation and the Nessyahu-Tedmor methods. The source terms play an important role and are discretized by the pointwise, upwind or mixed approaches. In the second part of this paper the above methods are verified by a set of test problems, covering all of three flow regimes: subcritical, supercritical, transcritical. The results show that the mixed approach of processing source terms is better than the pointwise one. The Roe approximation method with the mixed discretization of source terms is then applied for a preliminary evaluation of the Son La - Hoa Binh dam-break problem.

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Spa, Carlos, Otilio Rojas, and Josep de la Puente. "Comparison of expansion-based explicit time-integration schemes for acoustic wave propagation." GEOPHYSICS 85, no. 3 (2020): T165—T178. http://dx.doi.org/10.1190/geo2019-0462.1.

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We have developed a von Neumann stability and dispersion analysis of two time-integration techniques in the framework of Fourier pseudospectral (PS) discretizations of the second-order wave equation. The first technique is a rapid expansion method (REM) that uses Chebyshev matrix polynomials to approximate the continuous solution operator of the discrete wave equation. The second technique is a Lax-Wendroff method (LWM) that replaces time derivatives in the Taylor expansion of the solution wavefield with their equivalent spatial PS differentiations. In both time-integration schemes, each expansion term [Formula: see text] results in an extra application of the spatial differentiation operator; thus, both methods are similar in terms of their implementation and the freedom to arbitrarily increase accuracy by using more expansion terms. Nevertheless, their limiting Courant-Friedrichs-Lewy stability number [Formula: see text] and dispersion inaccuracies behave differently as [Formula: see text] varies. We establish the [Formula: see text] bounds for both methods in cases of practical use, [Formula: see text], and we confirm the results by numerical simulations. For both schemes, we explore the dispersion dependence on modeling parameters [Formula: see text] and [Formula: see text] on the wavenumber domain, through a new error metric. This norm weights errors by the source spectrum to adequately measure the accuracy differences. Then, we compare the theoretical computational costs of LWM and REM simulations to attain the same accuracy target by using the efficiency metric [Formula: see text]. In particular, we find optimal [Formula: see text] pairs that ensure a certain accuracy at a minimal computational cost. We also extend our dispersion analysis to heterogeneous media and find the LWM accuracy to be significantly better for representative [Formula: see text] values. Moreover, we perform 2D wave simulations on the SEG/EAGE Salt Model, in which larger REM inaccuracies are clearly observed on waveform comparisons in the range [Formula: see text].

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Gama, Italon Rilson Vicente, André Luiz Andrade Simões, Harry Edmar Schulz, and Rodrigo De Melo Porto. "CÓDIGO LIVRE PARA SOLUÇÃO NUMÉRICA DAS EQUAÇÕES DE SAINT-VENANT EM CANAIS TRAPEZOIDAIS ASSIMÉTRICOS." Revista Eletrônica de Gestão e Tecnologias Ambientais 8, no. 2 (2020): 145. http://dx.doi.org/10.9771/gesta.v8i2.38913.

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Ondas de cheia em canais e ondas produzidas por manobras em comportas são alguns fenômenos simulados com as equações de Saint-Venant em aplicações de engenharia. Um novo código foi desenvolvido para a solução dessas equações aplicadas a um canal trapezoidal assimétrico, empregando o método de volumes finitos de Lax e Friedrichs. Foi adotada uma linguagem de programação reconhecida por um software livre. Três testes numéricos foram realizados. O primeiro, correspondente à passagem de uma onda de cheia em um canal retangular, apresentou aderência aos resultados obtidos com a solução calculada através do método implícito de Preissmann, com desvio relativo máximo de 1,4% para a velocidade e de 0,81% para a altura de escoamento. O segundo teste resolveu o escoamento em um canal de fundo variado que induz à formação de um ressalto hidráulico. As comparações dos presentes resultados com aqueles de simulações publicadas recentemente resultaram em um desvio máximo de 2,3% para as alturas de escoamento, a montante e a jusante do ressalto hidráulico. Para as posições médias do ressalto hidráulico, o desvio foi de 2,4%. Na terceira comparação, simulou-se um ressalto hidráulico em um canal trapezoidal assimétrico de forte declividade, tendo sido encontrada uma solução com desvios relativos menores que 1% para os escoamentos a montante e a jusante do ressalto, quando comparados aos resultados calculados com o método de MacCormack. A posição média do ressalto nesta terceira comparação apresentou um desvio de 5,5% em relação aos resultados anteriores. Os desvios calculados indicam que o código desenvolvido é capaz de resolver escoamentos variáveis em canais com e sem a formação de ressaltos hidráulicos. Este é um resultado de cunho prático, pois mostra que códigos livres podem ser usados na prática da hidráulica em geometrias não-convencionais. OPEN SOURCE FOR NUMERICAL SOLUTION OF SAINT-VENAN EQUATIONS IN ASYMMETRIC TRAPEZOIDAL OPEN-CHANNELSFlood waves in channels, positive waves produced when operating floodgates, and the hydraulic jump are some phenomena simulated with the Saint-Venant equations in practical engineering applications. A new code was developed to solve these equations applied to an asymmetric trapezoidal channel using the Lax-Friedrichs finite volumes method. A programming language recognized by a free software was used. Three numerical tests were performed. The first, corresponding to the passage of a flood wave in a rectangular channel, showed adherence to results of the solution calculated using the Preissmann implicit method, presenting a maximum relative deviation of 1.4% for the speed and 0.81% for the flow height. The second test solved the flow in a channel with a variable bed that induces the formation of a hydraulic jump. Comparisons of the present results with those of recently published simulations produced a maximum deviation of 2.3% for the flow heights, upstream and downstream of the hydraulic jump. For the mean positions of the hydraulic jump the deviation was 2.4%. In the third comparison a hydraulic jump was simulated in an asymmetric trapezoidal channel with a strong slope, obtaining a solution with relative deviations less than 1% for flows upstream downstream of the jump, when compared to the results calculated with the MacCormack method. The average position of the jump in this third comparison showed a deviation of 5.5% in relation to the former results. The calculated deviations indicate that the developed code is capable of solving variable flows in channels with and without the formation of hydraulic jumps. This is a practical result, because it shows that open codes can be used in the practice of hydraulics in nonconventional geometries.

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Zhou, Xiaole, Haiqiang Lan, Ling Chen, et al. "An iterative factored topography-dependent eikonal solver for anisotropic media." GEOPHYSICS 86, no. 5 (2021): U121—U134. http://dx.doi.org/10.1190/geo2020-0662.1.

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Accurate and efficient eikonal solvers for heterogeneous media play an important role in many areas of seismology, such as seismic tomography, migration, and earthquake localization. Incorporating seismic anisotropy and complex topography remain a computational challenge for finite-difference eikonal solvers. In recent years, the topography-dependent eikonal equation (TDEE) has been proposed as an effective way to calculate seismic traveltimes for isotropic and anisotropic media with irregular topography. However, the Lax-Friedrichs sweeping method used in previous studies to approximate the viscosity solution of TDEE for anisotropic media is more dissipative and needs a much higher number of iterations to converge. In addition, the TDEE solution for the initial point source has an upwind source singularity, which makes all TDEE solvers, even the high-order ones, exhibit polluted convergence and relatively large errors that propagate from the point source to the entire computational domain. To solve these problems, we have formulated the factored topography-dependent anisotropic eikonal (FTDAE) equation in tilted transversely isotropic (TTI) media using the factorization principle. Then, the resulting quartic equation can be numerically solved by using a fixed-point iteration technique based on the simpler elliptical anisotropic eikonal (EAE) equation with a high-order source term due to anelliptical anisotropy introduced by TTI media. At each iteration, the unknown traveltime in the EAE equation is factored into two functions: One of the functions is specified analytically to capture the source singularity, such that the unknown factor is differentiable in the source neighborhood and could be solved by the fast sweeping method. Numerical examples indicate that our FTDAE solver can treat source singularity successfully and achieve high accuracy after just a few iterations, independently of the mesh size, which could provide a more efficient and robust tool for traveltime calculation in the presence of seismic anisotropy and complex surfaces.

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Zhalnin, Ruslan V., Victor F. Masyagin, Elizaveta E. Peskova, and Vladimir F. Tishkin. "Modeling the Flow of Multicomponent Reactive Gas on Unstructured Grids." Engineering Technologies and Systems 30, no. 1 (2020): 162–75. http://dx.doi.org/10.15507/2658-4123.030.202001.162-175.

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Introduction. The article deals with mathematical modeling of the subsonic flow of a multicomponent reactive mixture in a flowing chemical reactor. The numerical algorithm is based on the finite volume method; the calculation is performed on unstructured triangular grids using the Message Passing Interface parallel computing technology. Materials and Methods. To describe the flows under studying, the Navier–Stokes equations are used in the approximation for low Mach numbers. To solve these equations, the finite volume method on unstructured triangular grids is used. The study uses a splitting scheme for physical processes, namely, the chemical kinetics equations responsible for the transformations of substances are first solved, and then the equations describing the conservation laws of momentum and energy for each component of the gas mixture are solved. To find numerical flows through the edges of the grid elements, the Lax–Friedrichs–Rusanov scheme is used. To solve the equations of chemical kinetics, a compact algorithm proposed by the team led by N.N. Kalitkin is used. The METIS library is used to divide the grid into connected subdomains with an approximately equal number of cells. To organize parallel computing, Message Passing Interface technology is used. Results. The article presents a numerical algorithm for studying multicomponent gas flows on unstructured triangular grids taking into account viscosity, diffusion, thermal conductivity, and chemical reactions. As a result of the study, a numerical simulation of the flow of a subsonic multicomponent gas in a flowing chemical reactor was carried out using ethane pyrolysis as an example. Computational, known numerical solutions and experimental data of ethane pyrolysis in a flowing reactor are compared. Discussion and Conclusion. The numerical results on the conversion of the initial gas mixture are in good agreement with the known experimental data. The presented distribution patterns of the main components of the mixture and gas-dynamic parameters correspond to the flow pattern observed experimentally. Further work in this direction involves the modeling of subsonic gas flows on unstructured tetrahedral meshes using algorithms of higher accuracy for a more accurate study of ongoing processes.

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Cameron, Maria, Sergey Fomel, and James Sethian. "Time-to-depth conversion and seismic velocity estimation using time-migration velocity." GEOPHYSICS 73, no. 5 (2008): VE205—VE210. http://dx.doi.org/10.1190/1.2967501.

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The objective was to build an efficient algorithm (1) to estimate seismic velocity from time-migration velocity, and (2) to convert time-migrated images to depth. We established theoretical relations between the time-migration velocity and seismic velocity in two and three dimensions using paraxial ray-tracing theory. The relation in two dimensions implies that the conventional Dix velocity is the ratio of the interval seismic velocity and the geometric spreading of image rays. We formulated an inverse problem of finding seismic velocity from the Dix velocity and developed a numerical procedure for solving it. The procedure consists of two steps: (1) computation of the geometric spreading of image rays and the true seismic velocity in time-domain coordinates from the Dix velocity; (2) conversion of the true seismic velocity from the time domain to the depth domain and computation of the transition matrices from time-domain coordinates todepth. For step 1, we derived a partial differential equation (PDE) in two and three dimensions relating the Dix velocity and the geometric spreading of image rays to be found. This is a nonlinear elliptic PDE. The physical setting allows us to pose a Cauchy problem for it. This problem is ill posed, but we can solve it numerically in two ways on the required interval of time, if it is sufficiently short. One way is a finite-difference scheme inspired by the Lax-Friedrichs method. The second way is a spectral Chebyshev method. For step 2, we developed an efficient Dijkstra-like solver motivated by Sethian’s fast marching method. We tested numerical procedures on a synthetic data example and applied them to a field data example. We demonstrated that the algorithms produce a significantly more accurate estimate of seismic velocity than the conventional Dix inversion. This velocity estimate can be used as a reasonable first guess in building velocity models for depth imaging.

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Mwalimo, Delina Mshai, Mary Wainaina, and Winnie Kaluki. "Mixed Vehicular Traffic Flow Model on an Inclined Multilane Road." International Journal of Innovative Science and Research Technology 5, no. 7 (2020): 331–42. http://dx.doi.org/10.38124/ijisrt20jul276.

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This study outlines the Kerner’s 3 phase traffic flow theory, which states that traffic flow occurs in three phases and these are free flow, synchronized flow and wide moving jam phase. A macroscopic traffic model that is factoring road inclination is developed and its features discussed. By construction of the solution to the Rienmann problem, the model is written in conservative form and solved numerically. Using the Lax-Friedrichs method and going ahead to simulate traffic flow on an inclined multi lane road. The dynamics of traffic flow involving cars(fast moving) and trucks(slow moving) on a multi-lane inclined road is studied. Generally, trucks move slower than cars and their speed is significantly reduced when they are moving uphill on an in- clined road, which leads to emergence of a moving bottleneck. If the inclined road is multi-lane then the cars will tend to change lanes with the aim of overtaking the slow moving bottleneck to achieve free flow. The moving bottleneck and lanechange ma- noeuvres affect the dynamics of flow of traffic on the multi-lane road, leading to traffic phase transitions between free flow (F) and synchronised flow(S). Therefore, in order to adequately describe this kind of traffic flow, a model should incorporate the effect of road inclination. This study proposes to account for the road inclination through the fundamental diagram, which relates traffic flow rate to traffic density and ultimately through the anticipation term in the velocity dynamics equation of macroscopic traffic flow model. The features of this model shows how the moving bottleneck and an incline multilane road affects traffic transistions from Free flow(F) to Synchronised flow(S). For a better traffic management and control, proper understanding of traffic congestion is needed. This will help road designers and traffic engineers to verify whether traffic properties and characteristics such as speed(velocity), density and flow among others determines the effectiveness of traffic flow.

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Dallakyan, Gurgen. "Numerical Simulations for Chemotaxis Models." Biomath Communications 6, no. 1 (2019): 16. http://dx.doi.org/10.11145/bmc.2019.04.277.

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In the paper, we study the usage of numerical methods in solution of mathematical models of biological problems. More specifically, Keller-Segel type chemotaxis models are discussed, their numerical solutions by sweep and Lax-Friedrichs methods are obtained and interpreted biologically.

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Bodnár, Tomáš, Philippe Fraunié, and Karel Kozel. "MODIFIED EQUATION FOR A CLASS OF EXPLICIT AND IMPLICIT SCHEMES SOLVING ONE-DIMENSIONAL ADVECTION PROBLEM." Acta Polytechnica 61, SI (2021): 49–58. http://dx.doi.org/10.14311/ap.2021.61.0049.

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This paper presents the general modified equation for a family of finite-difference schemes solving one-dimensional advection equation. The whole family of explicit and implicit schemes working at two time-levels and having three point spatial support is considered. Some of the classical schemes (upwind, Lax-Friedrichs, Lax-Wendroff) are discussed as examples, showing the possible implications arising from the modified equation to the properties of the considered numerical methods.

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Coppo, Marco, Claudio Dongiovanni, and Claudio Negri. "Numerical Analysis and Experimental Investigation of a Common Rail-Type Diesel Injector." Journal of Engineering for Gas Turbines and Power 126, no. 4 (2004): 874–85. http://dx.doi.org/10.1115/1.1787502.

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A production common rail-type injector has been investigated via numerical simulation and experimentation. The functioning principle of the injector has been carefully analyzed so as to obtain a mathematical model of the device. A zero-dimensional approach has been used for modeling the injector, thus considering the variables as function of time only. The analysis of the hydraulic part of the injector resulted in the definition of an equivalent hydraulic scheme, on which basis both the equations of continuity in chambers and flow through nozzles were written. The connecting pipe between common rail and injector, as well as the injector internal line, were modeled according to a one-dimensional approach. The moving mechanical components of the injector, such as needle, pressure rod, and control valve have been modeled using the mass-spring-damper scheme, thus obtaining the equation governing their motion. An electromagnetic model of the control valve solenoid has also been realized, in order to work out the attraction force on the anchor, generated by the electric current when flowing into its coil. The model obtained has been implemented using the MATLAB® toolbox SIMULINK®; the ordinary differential equations were solved by means of an implicit scheme of the second-order accuracy, suitable for problems with high level of stiffness, while the partial differential equations were integrated using the finite-difference Lax-Friedrichs method. The experimental investigation on the common-rail injection system was performed on a test bench at some standard test conditions. Electric current flowing through the injector coil, oil pressure in the common rail and at the injector inlet, injection rate, needle lift, and control valve lift were gauged and recorded during several injection phases. The mean reflux-flow rate and the mean quantity of fuel injected per stroke were also measured. Temperature and pressure of the feeding oil as well as pressure in the rail were continuously controlled during the experimental test. The numerical and experimental results were compared. Afterwards, the model was used to investigate the effect of control volume feeding and discharge holes and of their inlet fillet, as well as the effect of the control volume capacity, on the injector performance.

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Rider, W. J., and R. B. Lowrie. "The use of classical Lax-Friedrichs Riemann solvers with discontinuous Galerkin methods." International Journal for Numerical Methods in Fluids 40, no. 3-4 (2002): 479–86. http://dx.doi.org/10.1002/fld.334.

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Chen, Weitao, Ching-Shan Chou, and Chiu-Yen Kao. "Lax–Friedrichs fast sweeping methods for steady state problems for hyperbolic conservation laws." Journal of Computational Physics 234 (February 2013): 452–71. http://dx.doi.org/10.1016/j.jcp.2012.10.008.

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Coelho, R. M. L., P. L. C. Lage, and A. Silva Telles. "A comparison of hyperbolic solvers II: ausm-type and Hybrid Lax-Wendroff-Lax-Friedrichs methods for two-phase flows." Brazilian Journal of Chemical Engineering 27, no. 1 (2010): 153–71. http://dx.doi.org/10.1590/s0104-66322010000100014.

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CHEN, GUI-QIANG, and ELEUTERIO F. TORO. "CENTERED DIFFERENCE SCHEMES FOR NONLINEAR HYPERBOLIC EQUATIONS." Journal of Hyperbolic Differential Equations 01, no. 03 (2004): 531–66. http://dx.doi.org/10.1142/s0219891604000202.

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A hierarchy of centered (non-upwind) difference schemes is identified for solving hyperbolic equations. The bottom of the hierarchy is the classical Lax–Friedrichs scheme, which is the least accurate in computation, and the top of the hierarchy is the FORCE scheme, which is the optimal scheme in the family. The FORCE scheme is optimal in the sense that it is monotone, has the optimal stability condition for explicit methods, and has the smallest numerical viscosity. It is shown that the FORCE scheme is consistent with the Lax entropy inequality, that is, the limit functions of the FORCE approximate solutions are entropy solutions. The convergence of the FORCE scheme is also established for the isentropic Euler equations and the shallow water equations. Some related centered difference schemes are also surveyed and discussed.

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Chen, Weitao, Ching-Shan Chou, and Chiu-Yen Kao. "Lax–Friedrichs Multigrid Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws." Journal of Scientific Computing 64, no. 3 (2015): 591–618. http://dx.doi.org/10.1007/s10915-015-0006-7.

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Li, Jia, Dazhi Zhang, Xiong Meng, Boying Wu, and Qiang Zhang. "Discontinuous Galerkin Methods for Nonlinear Scalar Conservation Laws: Generalized Local Lax--Friedrichs Numerical Fluxes." SIAM Journal on Numerical Analysis 58, no. 1 (2020): 1–20. http://dx.doi.org/10.1137/19m1243798.

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Wang, Dean, and Tseelmaa Byambaakhuu. "High-Order Lax-Friedrichs WENO Fast Sweeping Methods for the SN Neutron Transport Equation." Nuclear Science and Engineering 193, no. 9 (2019): 982–90. http://dx.doi.org/10.1080/00295639.2019.1582316.

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Chen, Weitao, Ching-Shan Chou, and Chiu-Yen Kao. "Erratum to: Lax–Friedrichs Multigrid Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws." Journal of Scientific Computing 64, no. 3 (2015): 619. http://dx.doi.org/10.1007/s10915-015-0025-4.

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Koroche, Kedir Aliyi. "Numerical Solution of In-Viscid Burger Equation in the Application of Physical Phenomena: The Comparison between Three Numerical Methods." International Journal of Mathematics and Mathematical Sciences 2022 (March 29, 2022): 1–11. http://dx.doi.org/10.1155/2022/8613490.

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In this paper, upwind approach, Lax–Friedrichs, and Lax–Wendroff schemes are applied for working solution of In-thick Burger equation in the application of physical phenomena and comparing their error norms. First, the given solution sphere is discretized by using an invariant discretization grid point. Next, by using Taylor series expansion, we gain discretized nonlinear difference scheme of given model problem. By rearranging this scheme, we gain three proposed schemes. To verify validity and applicability of proposed techniques, one model illustration with subordinated to three different original conditions that satisfy entropy condition are considered, and solved it at each specific interior grid points of solution interval, by applying all of the techniques. The stability and convergent analysis of present three techniques are also worked by supporting both theoretical and numerical fine statements. The accuracy of present techniques has been measured in the sense of average absolute error, root mean square error, and maximum absolute error norms. Comparisons of numerical gets crimes attained by these three methods are presented in table. Physical behaviors of numerical results are also presented in terms of graphs. As we can see from numerical results given in both tables and graphs, the approximate solution is good agreement with exact solutions. Therefore, the present systems approaches are relatively effective and virtually well suited to approximate the solution of in-viscous Burger equation.

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