Journal articles: 'Lax-Wendroff scheme'

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Nigar, Sultana Laek Sazzad Andallah. "Stability Analysis Of First And Second Order Explicit Finite Difference Scheme of Advection-Diffusion Equation." Multicultural Education 8, no. 2 (2022): 52. https://doi.org/10.5281/zenodo.5973240.

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<em>This article studies two first order schemes, FTBSCS and FTCSCS and propose a second order Lax-Wendroff type scheme for the numerical solutions of advection-diffusion equation (ADE) as an initial and boundary value problem. In previous Lax-Wendroff scheme was introduced only for hyperbolic partial differential equation (PDE) but here a new second order Lax-Wendroff type explicit finite difference scheme is proposed for parabolic ADE. For proposed second order Lax-Wendorff type scheme of ADE, we discretise the first order terms of ADE in second order like Lax-Wendorff scheme for hyperbolic partial differential equation. We perform stability analysis of these numerical schemes and determine the condition of stability in terms of temporal and spatial step sizes, advection co-efficient and diffusion co-efficient. The stability conditions of these schemes lead to determine the efficiency of these schemes in terms of the time step restrictions. Finally, we compare these schemes in terms of stability condition and efficiency as well.</em>

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Laek Sazzad Andallah, Nigar Sultana,. "Investigation of Water Pollution in the River with Second-Order Explicit Finite Difference Scheme of Advection-Diffusion Equation and First-Order Explicit Finite Difference Scheme of Advection-Diffusion Equation." Mathematical Statistician and Engineering Applications 71, no. 2 (2022): 12–27. http://dx.doi.org/10.17762/msea.v71i2.62.

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The perseverance of this research article is to investigate water pollution in rivers with a second-order explicit finite difference scheme of advection-diffusion equation (ADE) and a first-order explicit finite difference scheme of ADE. For investigation, two numerical schemes exploit here FTCSCS and second-order Lax-Wendroff type of ADE which is our new proposed one. In earlier Lax-Wendroff, type scheme existed only for hyperbolic partial differential equation (PDE), here a new second-order Lax-Wendroff type scheme is proposed for parabolic PDE and in addition assist to investigate water pollution with an expectation of better yield compared to the existing one. We implement numerical schemes to estimate the pollutant in water at different times and different points of water bodies. We investigate the numerical behaviour of water pollution by implementing the explicit centred difference scheme (FTCSCS) for advection-diffusion and for our proposed second-order Lax-Wendroff type scheme. Our computational result verifies the qualitative behavior of the solution of ADE for various considerations of the parameters.

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Dong, Haoyu, Changna Lu, and Hongwei Yang. "The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations." Mathematics 6, no. 10 (2018): 211. http://dx.doi.org/10.3390/math6100211.

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We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

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Lu, Changna, Luoyan Xie, and Hongwei Yang. "The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography." Mathematical Problems in Engineering 2018 (2018): 1–15. http://dx.doi.org/10.1155/2018/2652367.

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A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.

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Fadhli Ahmad, Mohammad, Mohd Sofiyan Suliman, and . "Deficiency of finite difference methods for capturing shock waves and wave propagation over uneven bottom seabed." International Journal of Engineering & Technology 7, no. 3.28 (2018): 97. http://dx.doi.org/10.14419/ijet.v7i3.28.20977.

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The implementation of finite difference method is used to solve shallow water equations under the extreme conditions. The cases such as dam break and wave propagation over uneven bottom seabed are selected to test the ordinary schemes of Lax-Friederichs and Lax-Wendroff numerical schemes. The test cases include the source term for wave propagation and exclude the source term for dam break. The main aim of this paper is to revisit the application of Lax-Friederichs and Lax-Wendroff numerical schemes at simulating dam break and wave propagation over uneven bottom seabed. For the case of the dam break, the two steps of Lax-Friederichs scheme produce non-oscillation numerical results, however, suffering from some of dissipation. Moreover, the two steps of Lax-Wendroff scheme suffers a very bad oscillation. It seems that these numerical schemes cannot solve the problem at discontinuities which leads to oscillation and dissipation. For wave propagation case, those numerical schemes produce inaccurate information of free surface and velocity due to the uneven seabed profile. Therefore, finite difference is unable to model shallow water equations under uneven bottom seabed with high accuracy compared to the analytical solution.

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Chen, Jing-Bo. "High-order time discretizations in seismic modeling." GEOPHYSICS 72, no. 5 (2007): SM115—SM122. http://dx.doi.org/10.1190/1.2750424.

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Seismic modeling plays an important role in explor-ation geophysics. High-order modeling schemes are in demand for practical reasons. In this context, I present three kinds of high-order time discretizations: Lax-Wendroff methods, Nyström methods, and splitting methods. Lax-Wendroff methods are based on the Taylor expansion and the replacement of high-order temporal derivatives by spatial derivatives, Nyström methods are simplified Runge-Kutta algorithms, and splitting methods comprise substeps for one-step computation. Based on these methods, three schemes with third-order and fourth-order accuracy in time and pseudospectral discretizations in space are presented. I also compare their accuracy, stability, and computational complexity, and discuss advantages and shortcomings of these algorithms. Numerical experiments show that the fourth-order Lax-Wendroff scheme is more efficient for short-time simulations while the fourth-order Nyström scheme and the third-order splitting scheme are more efficient for long-term computations.

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Appadu, A. R. "Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes." Journal of Applied Mathematics 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/734374.

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Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted byhandk, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advection-diffusion equation at some values ofkandh. Two optimisation techniques are then implemented to find the optimal values ofkwhenh=0.02for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.

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Ren, Zhiming, Qianzong Bao, and Bingluo Gu. "Time-dispersion correction for arbitrary even-order Lax-Wendroff methods and the application on full-waveform inversion." GEOPHYSICS 86, no. 5 (2021): T361—T375. http://dx.doi.org/10.1190/geo2020-0934.1.

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A second-order accurate finite-difference (FD) approximation is commonly used to approximate the second-order time derivative of a wave equation. The second-order accurate FD scheme may introduce time dispersion in wavefield extrapolation. Lax-Wendroff methods can suppress such dispersion by replacing the high-order time FD error terms with space FD error-correcting terms. However, the time dispersion cannot be completely eliminated and the computation cost dramatically increases with increasing order of (temporal) accuracy. To mitigate the problem, we have extended the existing time-dispersion correction scheme for the second- or fourth-order Lax-Wendroff method to a scheme for arbitrary even-order methods, which uses the forward and inverse time-dispersion transform (FTDT and ITDT) to add and remove the time dispersion from synthetic data. We test the correction scheme using a homogeneous model and the Sigsbee2A model. The modeling examples suggest that the use of derived FTDT and ITDT pairs on high-order Lax-Wendroff methods can effectively remove time-dispersion errors from high-frequency waves while using longer time steps than allowed in low-order Lax-Wendroff methods. We investigate the influence of the time dispersion on full-waveform inversion (FWI) and show an antidispersion workflow. We apply the FTDT to source terms and recorded traces before inversion, resulting in the source and adjoint wavefields containing equal time dispersion from source-side wave propagation and the modeled and observed traces accumulating equal time dispersion from source- and receiver-side wave propagation. The inversion results reveal that the antidispersion workflow is capable of increasing the accuracy of FWI for arbitrary even-order Lax-Wendroff methods. In addition, the high-order method can obtain better inversion results compared to the second-order method with the same antidispersion workflow.

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Rahman, Md Mizanur, K. Hasan, Zhiqian Sang, and Zing Ni. "Lax-Wendroff method for incompressible flow." Journal of Physics: Conference Series 2313, no. 1 (2022): 012002. http://dx.doi.org/10.1088/1742-6596/2313/1/012002.

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Abstract The Lax-Wendroff scheme is extended to incompressible fluid flow problems within the framework of artificial compressibility method (ACM) utilizing a “cell-centered finite-volume” Δ-approximation on a non-orthogonal non-staggered grid. An ACM induces a transformation between conservative and primitive variables; the physical relevance of ACM is to bring about “matrix preconditionings”, provoking density perturbations. The coupled algorithm is pressure-based; it benefits from enhanced accuracy and greater flexibility using the “monotone upstream-centered schemes for conservation laws (MUSCL)” approach. Numerical experiments in reference to “buoyancy-driven” flows with strong sources illustrate that the entire strategy augments overall damping capability and robustness adhering to the factored pseudo-time integration method. Conclusively, the associated limiter function has little influence on the high resolution for selected test cases.

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Bergmann, Tim, Joakim O. Blanch, Johan O. A. Robertsson, and Klaus Holliger. "A simplified Lax‐Wendroff correction for staggered‐grid FDTD modeling of electromagnetic wave propagation in frequency‐dependent media." GEOPHYSICS 64, no. 5 (1999): 1369–77. http://dx.doi.org/10.1190/1.1444642.

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The Lax‐Wendroff correction is an elegant method for increasing the accuracy and computational efficiency of finite‐difference time‐domain (FDTD) solutions of hyperbolic problems. However, the conventional approach leads to implicit solutions for staggered‐grid FDTD approximations of Maxwell’s equations with frequency‐dependent constitutive parameters. To overcome this problem, we propose an approximation that only retains the purely acoustic, i.e., lossless, terms of the Lax‐Wendroff correction. This modified Lax‐Wendroff correction is applied to an O(2, 4) accurate staggered‐grid FDTD approximation of Maxwell’s equations in the radar frequency range (≈10 MHz–10 GHz). The resulting pseudo-O(4, 4) scheme is explicit and computationally efficient and exhibits all the major numerical characteristics of an O(4, 4) accurate FDTD scheme, even for strongly attenuating and dispersive media. The numerical properties of our approach are constrained by classical numerical dispersion and von Neumann‐Routh stability analyses, verified by comparisons with pertinent 1-D analytical solutions and illustrated through 2-D simulations in a variety of surficial materials. Compared to the O(2, 4) scheme, the pseudo-O(4, 4) scheme requires 64% fewer grid points in two dimensions and 78% in three dimensions to achieve the same level of numerical accuracy, which results in large savings in core memory.

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Jejeniwa, Olaoluwa Ayodeji, Hagos Hailu Gidey, and Appanah Rao Appadu. "Numerical Modeling of Pollutant Transport: Results and Optimal Parameters." Symmetry 14, no. 12 (2022): 2616. http://dx.doi.org/10.3390/sym14122616.

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In this work, we used three finite difference schemes to solve 1D and 2D convective diffusion equations. The three methods are the Kowalic–Murty scheme, Lax–Wendroff scheme, and nonstandard finite difference (NSFD) scheme. We considered a total of four numerical experiments; in all of these cases, the initial conditions consisted of symmetrical profiles. We looked at cases when the advection velocity was much greater than the diffusion of the coefficient and cases when the coefficient of diffusion was much greater than the advection velocity. The dispersion analysis of the three methods was studied for one of the cases and the optimal value of the time step size k, minimizing the dispersion error at a given value of the spatial step size. From our findings, we conclude that Lax–Wendroff is the most efficient scheme for all four cases. We also show that the optimal value of k computed by minimizing the dispersion error at a given value of a spacial step size gave the lowest l2 and l∞ errors.

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Galaguz, Yuri P. "REALIZATION OF THE TVD-SCHEME FOR A NUMERICAL SOLUTION OF THE FILTRATION PROBLEM." International Journal for Computational Civil and Structural Engineering 13, no. 2 (2017): 93–102. http://dx.doi.org/10.22337/2587-9618-2017-13-2-93-102.

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A numerical solution of one-dimensional filtration problem for suspension flow in a porous medium is considered. The TVD-scheme with different functions-delimiters is obtained. The TVD-scheme is compared with the counter-current scheme and the Lax-Wendroff scheme. Diagrams of suspended particles concentrations are shown.

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Saxena, Parul, Vinay Saxena, and Raju Prasad. "Numerical Investigation of 1D Burgers' equation using Lax-Friedrichs and Lax-Wendroff schemes." Anthology The Research 8, no. 11 (2024): E 26 — E 35. https://doi.org/10.5281/zenodo.10907849.

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This paper has been published in Peer-reviewed International Journal "Anthology The Research"                URL : https://www.socialresearchfoundation.com/new/publish-journal.php?editID=8663 Publisher : Social Research Foundation, Kanpur (SRF International)                  Abstract :  This research investigates the efficacy of the Lax-Friedrichs and Lax-Wendroff schemes in solving the 1D Burgers' equation, emphasizing the impact of varying viscosity coefficients. With a focus on numerical stability, accuracy, and computational efficiency, the two schemes are implemented and compared through color-coded visualizations at different time steps. The study addresses the need for efficient and accurate numerical tools to understand the complexities of fluid flow dynamics. Results showcase the schemes' performance under various viscosity conditions, offering insights into their strengths and weaknesses. The research contributes valuable information for fluid dynamics simulations, aiding researchers and practitioners in selecting suitable numerical methods for specific applications.

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Nikolopoulos, C. V. "Numerical Solution of a Nonlocal Problem Modelling Ohmic Heating of Foods." Computational Methods in Applied Mathematics 9, no. 4 (2009): 391–411. http://dx.doi.org/10.2478/cmam-2009-0025.

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Abstract An upwind and a Lax-Wendroff scheme are introduced for the solution of a one-dimensional non-local problem modelling ohmic heating of foods. The schemes are studied regarding their consistency, stability, and the rate of convergence for the cases that the problem attains a global solution in time. A high resolution scheme is also introduced and it is shown that it is total-variation-stable. Finally some numerical experiments are presented in support of the theoretical results.

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FRICKE, J. ROBERT. "QUASI-LINEAR ELASTODYNAMIC EQUATIONS FOR FINITE DIFFERENCE SOLUTIONS IN DISCONTINUOUS MEDIA." Journal of Computational Acoustics 01, no. 03 (1993): 303–20. http://dx.doi.org/10.1142/s0218396x93000160.

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The linear elastodynamic equations are ill-posed for models which contain high contrast density discontinuities. This paper presents a quasi-linear superset of the linear equations that is well-posed for this situation. The extended system contains a conservation of mass equation and a quasi-linear convective term in the momentum equation. Density, momentum, and stress are the field variables in the quasi-linear system, which is cast in a first order form. Using a Lax–Wendroff finite difference approximation, the utility of the quasi-linear system is demonstrated by modeling underwater acoustic scattering from a truncated ice sheet. The model contains air, ice, and water with a density contrast between air and ice or water of O(103). Superlinear convergence of the Lax–Wendroff scheme is demonstrated for his heterogeneous medium problem.

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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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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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Santosa, Fadil, and Yih‐Hsing Pao. "Accuracy of a Lax–Wendroff scheme for the wave equation." Journal of the Acoustical Society of America 80, no. 5 (1986): 1429–37. http://dx.doi.org/10.1121/1.394398.

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Fridrich, David, Richard Liska, Ivan Tarant, Pavel Váchal, and Burton Wendroff. "CELL-CENTERED LAGRANGIAN LAX-WENDROFF HLL HYBRID SCHEME ON UNSTRUCTURED MESHES." Acta Polytechnica 61, SI (2021): 68–76. http://dx.doi.org/10.14311/ap.2021.61.0068.

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We have recently introduced a new cell-centered Lax-Wendroff HLL hybrid scheme for Lagrangian hydrodynamics [Fridrich et al. J. Comp. Phys. 326 (2016) 878-892] with results presented only on logical rectangular quadrilateral meshes. In this study we present an improved version on unstructured meshes, including uniform triangular and hexagonal meshes and non-uniform triangular and polygonal meshes. The performance of the scheme is verified on Noh and Sedov problems and its second-order convergence is verified on a smooth expansion test.Finally the choice of the scalar parameter controlling the amount of added artificial dissipation is studied.

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Potapov, I. I., and P. S. Timosh. "ON THE USE OF THE CENTRAL DIFFERENCE SCHEME FOR SOLVING THE PROBLEM OF GAS DYNAMICS." Informatika i sistemy upravleniya, no. 2 (2021): 17–22. http://dx.doi.org/10.22250/isu.2021.68.17-22.

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The paper proposes a method for solving the problem of gas dynamics, implemented on the basis of a central difference scheme, the stability of which is achieved by performing a correction of the calcu-lated flows. It is shown that when solving the problem of discontinuity decay, the proposed method is stable, comparable in accuracy with the McCormack and Lax – Wendroff methods and surpasses them in performance.

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Fridrich, David, Richard Liska, and Burton Wendroff. "Cell-centered Lagrangian Lax-Wendroff HLL hybrid scheme in cylindrical geometry." Journal of Computational Physics 417 (September 2020): 109605. http://dx.doi.org/10.1016/j.jcp.2020.109605.

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Collins, J. B., Don Estep, and Simon Tavener. "A posteriori error estimation for the Lax–Wendroff finite difference scheme." Journal of Computational and Applied Mathematics 263 (June 2014): 299–311. http://dx.doi.org/10.1016/j.cam.2013.12.035.

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Benoit, Antoine, and Jean-François Coulombel. "The Neumann boundary condition for the two-dimensional Lax–Wendroff scheme." Communications in Mathematical Sciences 21, no. 8 (2023): 2051–82. http://dx.doi.org/10.4310/cms.2023.v21.n8.a1.

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Pei, Yanrong, Haifang Jian, and Wenchang Li. "An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation." Applied Sciences 13, no. 21 (2023): 11713. http://dx.doi.org/10.3390/app132111713.

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The stability and accuracy of explicit high-order finite difference (HOFD) algorithms have been research hotspots in different fields. To improve the stability and accuracy of the HOFD algorithms in thermal simulations, we present a Lax-Wendroff high-order finite difference (LHOFD) algorithm to solve the 2D transient heat transfer equation in this paper and develop an improved LHOFD (IHOFD) algorithm to improve the stability of the LHOFD algorithm. The formulas of the general high-order central FD (HOCFD) coefficients and the truncation error coefficient as well as the high-order non-central FD (HONFD) coefficients and the truncation error coefficient of the fourth-order spatial derivative are derived concisely in a different way. Furthermore, a unified analytical formula of the general HOCFD and HONFD coefficients, which can calculate the spatial derivative of any integer order, is derived. A new strategy of combination with the HOCFD and HONFD approximations under the same high-order accuracy as the internal computational domain is proposed to calculate the mixed derivatives of the boundary domains with high accuracy, no additional computational cost, and easy implementation. Then, the accuracy analysis, stability analysis, and comparative analysis of numerical simulation results obtained by the LHOFD and IHOFD algorithms with the exact solution show the correctness and validity of the proposed algorithms and their stability formulas, and the advantages of the proposed algorithms. The proposed algorithms are valid under both symmetric and asymmetric boundary conditions. The stability factor of the LHOFD algorithm is slightly higher than that of the conventional algorithm. The stability factor of the IHOFD algorithm is twice that of the conventional algorithm, and the maximum absolute error of the thermal simulation is within 0.015 (°C).

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Ding, Lijuan. "Accuracy of Lax-Wendroff scheme for discontinuous solutions of convection equations." Chinese Science Bulletin 42, no. 24 (1997): 2047–51. http://dx.doi.org/10.1007/bf02882942.

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Peng, Shuang, Songze Chen, Hong Liang, and Chuang Zhang. "Semi-implicit Lax-Wendroff kinetic scheme for multi-scale phonon transport." Computers & Mathematics with Applications 187 (June 2025): 72–84. https://doi.org/10.1016/j.camwa.2025.03.019.

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Чижонков, Е. В. "On second-order accuracy schemes for modeling of plasma oscillations." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 1 (January 13, 2020): 115–28. http://dx.doi.org/10.26089/nummet.v21r110.

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Для моделирования колебаний холодной плазмы как в нерелятивистском случае, так и с учетом релятивизма предложены модификации классических разностных схем второго порядка точности: метода МакКормака и двухэтапного метода Лакса-Вендроффа. Ранее для подобных расчетов в эйлеровых переменных была известна только схема первого порядка точности. Для задачи о свободных плазменных колебаниях, инициированных коротким мощным лазерным импульсом, с целью тестирования представленных схем проведены численные эксперименты по сохранению энергии и других величин. Сделан вывод о достоверности численного анализа колебаний как на основе схемы МакКормака, так и на основе схемы Лакса-Вендроффа, однако для расчетов долгоживущих процессов первая схема более предпочтительна. Теоретическое исследование аппроксимации и устойчивости вместе с экспериментальным наблюдением за количественными характеристиками погрешности для наиболее чувствительных величин существенно повышает достоверность вычислений. Ключевые слова: численное моделирование, плазменные колебания, эффект опрокидывания, схемы МакКормака и Лакса-Вендроффа, порядок точности разностной схемы, законы сохранения. For modeling cold plasma oscillations in the non-relativistic and relativistic cases, some modifications of classical difference schemes of the second order of accuracy are proposed: the McCormack method and the two-stage Lax-Wendroff method. Previously, only the first-order accuracy scheme was known for calculations in Euler variables. For the problem of free plasma oscillations initiated by a short high-power laser pulse, the results of numerical experiments on energy conservation and other quantities were performed in order to test the proposed schemes. It is concluded that the numerical analysis of oscillations is reliable both for the McCormack scheme and for the Lax-Wendroff scheme however, for the calculation of long-lived processes, the first scheme is more preferable. The theoretical analysis of approximation and stability together with experimental observations of quantitative characteristics of errors for the most sensitive quantities significantly increases the reliability of calculations.

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Machalinska-Murawska, Justyna, and Michał Szydłowski. "Lax-Wendroff and McCormack Schemes for Numerical Simulation of Unsteady Gradually and Rapidly Varied Open Channel Flow." Archives of Hydro-Engineering and Environmental Mechanics 60, no. 1-4 (2014): 51–62. http://dx.doi.org/10.2478/heem-2013-0008.

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Abstract Two explicit schemes of the finite difference method are presented and analyzed in the paper. The applicability of the Lax-Wendroff and McCormack schemes for modeling unsteady rapidly and gradually varied open channel flow is investigated. For simulation of the transcritical flow the original and improved McCormack scheme is used. The schemes are used for numerical solution of one dimensional Saint-Venant equations describing free surface water flow. Two numerical simulations of flow with different hydraulic characteristics were performed - the first one for the extreme flow of the dam-break type and the second one for the simplified flood wave propagation problem. The computational results are compared to each other and to arbitrary solutions.

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Feng, Renzhong, and Zheng Wang. "Simple and High-Accurate Schemes for Hyperbolic Conservation Laws." Journal of Applied Mathematics 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/275425.

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The paper constructs a class of simple high-accurate schemes (SHA schemes) with third order approximation accuracy in both space and time to solve linear hyperbolic equations, using linear data reconstruction and Lax-Wendroff scheme. The schemes can be made even fourth order accurate with special choice of parameter. In order to avoid spurious oscillations in the vicinity of strong gradients, we make the SHA schemes total variation diminishing ones (TVD schemes for short) by setting flux limiter in their numerical fluxes and then extend these schemes to solve nonlinear Burgers’ equation and Euler equations. The numerical examples show that these schemes give high order of accuracy and high resolution results. The advantages of these schemes are their simplicity and high order of accuracy.

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Zhang, Yuangao, and Behrouz Tabarrok. "Modifications to the Lax?Wendroff scheme for hyperbolic systems with source terms." International Journal for Numerical Methods in Engineering 44, no. 1 (1999): 27–40. http://dx.doi.org/10.1002/(sici)1097-0207(19990110)44:1<27::aid-nme485>3.0.co;2-0.

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Abedian, Rooholah. "High-Order Semi-Discrete Central-Upwind Schemes with Lax–Wendroff-Type Time Discretizations for Hamilton–Jacobi Equations." Computational Methods in Applied Mathematics 18, no. 4 (2018): 559–80. http://dx.doi.org/10.1515/cmam-2017-0031.

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AbstractA new fifth-order, semi-discrete central-upwind scheme with a Lax–Wendroff time discretization procedure for solving Hamilton–Jacobi (HJ) equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. Unlike most of the commonly used high-order upwind schemes, the new scheme is formulated as a Godunov-type method. The new scheme is based on the flux Kurganov, Noelle and Petrova (KNP flux). The spatial discretization is based on a symmetrical weighted essentially non-oscillatory reconstruction of the derivative. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. Various numerical experiments are performed to demonstrate the accuracy and stability properties of the new method. As a result, comparing with other fifth-order schemes for HJ equations, the major advantage of the new scheme is more cost effective for certain problems while the new method exhibits smaller errors without any increase in the complexity of the computations.

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Zendrato, Nur Lely Hardianti, Asrini Chrysanti, Bagus Pramono Yakti, Mohammad Bagus Adityawan, Widyaningtias, and Yadi Suryadi. "Application of Finite Difference Schemes to 1D St. Venant for Simulating Weir Overflow." MATEC Web of Conferences 147 (2018): 03011. http://dx.doi.org/10.1051/matecconf/201814703011.

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Depth averaged equations are commonly used for modelling hydraulics problems. Nevertheless, the model may not be able to accurately assess the flow in the case of different flow regimes, such as hydraulic jump. The model requires appropriate numerical method or other numerical treatments in order to simulate the case accurately. A finite volume scheme with shock capturing may provide a good result, but it is time consuming as compared to the commonly used finite difference schemes. In this study, 1D St. Venant equation is solved using Artificial Viscosity Lax-Wendroff and Mac-Cormack with TVD filter schemes to simulate an experiment case of weir overflow. The case is chosen to test each scheme ability in simulating flow under different flow regimes. The simulation results are benchmarked to the observed experimental data from previous study. Additionally, to observe the scheme efficiency, the simulation time between the models are compared. Therefore, the most accurate and efficient scheme can be determined.

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OGATA, YOUICHI, TAKASHI YABE, KAZUNARI SHIBATA, and TAKAHIRO KUDOH. "EFFICIENT COMPUTATION OF MAGNETO-HYDRODYNAMIC PHENOMENA IN ASTROPHYSICS BY CCUP-MOCCT METHOD." International Journal of Computational Methods 01, no. 01 (2004): 201–25. http://dx.doi.org/10.1142/s021987620400006x.

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We proved that the CIP (Constrained Interpolation Profile/Cubic Interpolated Pseudoparticle)-MOCCT (Method of Characteristics/Constrained Transport) is capable of treating MHD phenomena using a mesh several times coarser than the conventional schemes. Comparing the present method with the modified Lax-Wendroff methods in the Parker instability in two and three dimensions, the superiority of the CIP scheme has been demonstrated from its low numerical phase error and damping rate. In addition, we employ the CIP-CUP (CIP-combined unified procedure/CCUP) method that solves the Poisson equation for pressure in order to make the present scheme more stable. This method is able to treat the pressure scale height having just one grid, which is five times coarser than the conventional scheme, as well as the magnetic reconnection, which has been described with a coarse grid.

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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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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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BOUSHABA, FARID, ELMILOUD CHAABELASRI, NAJIM SALHI, IMAD ELMAHI, FAYSSAL BENKHALDOUN, and ALISTAIR G. L. BORTHWICK. "A COMPARATIVE STUDY OF FINITE VOLUME AND FINITE ELEMENT ON SOME TRANSCRITICAL FREE SURFACE FLOW PROBLEMS." International Journal of Computational Methods 05, no. 03 (2008): 413–31. http://dx.doi.org/10.1142/s0219876208001522.

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This paper presents details of finite volume and finite element numerical models based on unstructured triangular meshes that are used to solve the two-dimensional nonlinear shallow water equations (SWEs). The finite volume scheme uses Roe's approximate Riemann solver to evaluate the convection terms. Second order accuracy is achieved by means of the MUSCL approach with MinMod and VanAlbada limiters. The finite element model utilizes the Lax–Wendroff two-step scheme, which is second-order in space and time. The models are validated and their relative performance compared for several benchmark problems, including a hydraulic jump, and flows in converging and converging–diverging channels.

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Mustafa, Muhammad I., Salim A. Messaoudi, and Mostafa Zahri. "Theoretical and computational results of a wave equation with variable exponent and time-dependent nonlinear damping." Arabian Journal of Mathematics 10, no. 2 (2021): 443–58. http://dx.doi.org/10.1007/s40065-021-00312-6.

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AbstractWe study the following wave equation $$u_{tt}-\Delta u+\alpha (t)\left| u_{t}\right| ^{m(\cdot )-2}u_{t}=0$$ u tt - Δ u + α ( t ) u t m ( · ) - 2 u t = 0 with a nonlinear damping having a variable exponent m(x) and a time-dependent coefficient $$\alpha (t)$$ α ( t ) . We use the multiplier method to establish energy decay results depending on both m and $$\alpha $$ α . We also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme.

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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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39

Lee, Wonwoong, Jae Jun Lee, and Jeong Ik Lee. "Analysis of nodalization uncertainty for nuclear system analysis code with Lax-Wendroff numerical scheme." Annals of Nuclear Energy 167 (March 2022): 108853. http://dx.doi.org/10.1016/j.anucene.2021.108853.

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Vafidis, A., F. Abramovici, and E. R. Kanasewich. "Elastic wave propagation using fully vectorized high order finite‐difference algorithms." GEOPHYSICS 57, no. 2 (1992): 218–32. http://dx.doi.org/10.1190/1.1443235.

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Two finite‐difference schemes for solving the elastic wave equation in heterogeneous two‐dimensional media are implemented on a vector computer. A modified Lax‐Wendroff scheme that is second‐order accurate both in time and space and is a version of the MacCormack scheme that is second‐order accurate in time and fourth‐order in space. The algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations. The technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences. The two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two‐layer model with a line source. A comparison of the results for an actual well‐to‐well experiment verifies the usefulness of the two‐dimensional approach in modeling the results.

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Иванов, Д. В., Г. М. Кобельков, М. А. Ложников, and А. Ф. Харисов. "A method of adaptive artificial viscosity for solving numerically the equations of a viscous heat-conducting compressible gas." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 1(55) (March 13, 2018): 51–62. http://dx.doi.org/10.26089/nummet.v19r105.

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Статья посвящена численному решению уравнений динамики вязкого сжимаемого теплопроводного газа на неструктурированных тетраэдальных сетках. Предложена комбинация методов МакКормака и Лакса-Вендроффа, которая позволяет провести приближенную монотонизацию разностной схемы с помощью введения адаптивной искусственной вязкости и метода "замороженных" коэффициентов. Результаты расчетов согласуются с натурными экспериментами. This paper is devoted to the numerical solution of the dynamics equations for a viscous heat-conducting compressible gas by the method of adaptive viscosity on unstructured tetrahedral meshes. A combination of the MacCormack method and the Lax-Wendroff method allows one to monotonize the difference scheme using the method of frozen coefficients. The numerical results are in good agreement with experimental data.

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Karakozova, Anastasia, and Sergey Kuznetsov. "Oscillating Nonlinear Acoustic Waves in a Mooney–Rivlin Rod." Applied Sciences 13, no. 18 (2023): 10037. http://dx.doi.org/10.3390/app131810037.

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Harmonic wave excitation in a semi-infinite incompressible hyperelastic 1D rod with the Mooney–Rivlin equation of state reveals the formation and propagation of the shock wave fronts arising between faster and slower moving parts of the initially harmonic wave. The observed shock wave fronts result in the collapse of the slower moving parts being absorbed by the faster parts; hence, to the attenuation of the kinetic and the elastic strain energy with the corresponding heat generation. Both geometrically and physically nonlinear equations of motion are solved by the explicit Lax–Wendroff numerical tine-integration scheme combined with the finite element approach for spatial discretization.

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Zhai, Qinglan, Song Zheng, and Lin Zheng. "A kinetic theory based thermal lattice Boltzmann equation model." International Journal of Modern Physics C 28, no. 04 (2017): 1750047. http://dx.doi.org/10.1142/s0129183117500474.

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A thermal lattice Boltzmann equation (LBE) model within the framework of double distribution function (DDF) method is proposed from the continuous DDF Boltzmann equation, which has a clear physical significance. Since the discrete velocity set in present LBE model is not space filled, a Lax–Wendroff scheme is applied to solve the evolution equations by which the spatial interpolation of two distribution functions is overcome. To validate the model, some classical numerical tests include thermal Couette flow and natural convection flow are simulated, and the results agree well with the analytic solutions and other numerical results, which showed that the present model had the ability to describe the thermal fluid flow phenomena.

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Kuznetsov, Sergey V. "Shock Wave Formation and Cloaking in Hyperelastic Rods." Applied Sciences 13, no. 8 (2023): 4740. http://dx.doi.org/10.3390/app13084740.

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The analysis of propagating an initially harmonic acoustic pulse in a semi-infinite hyperelastic rod obeying the Yeoh strain energy potential reveals attenuation with distance of the wave amplitudes caused by the elastic energy dissipation due to forming and propagation of the shock wave fronts and heat production. The observed attenuation of harmonic waves results in a broadband cloaking of fairly remote regions. The analysis is based on solving a nonlinear equation of motion by an explicit Lax–Wendroff time-difference scheme combined with the finite element discretization in the spatial domain. The revealing phenomena are applicable to studies of acoustic wave propagation in various elastomeric rubberlike materials modeled by the Yeoh hyperelastic potentials.

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Cichy, D. M., B. Fikus, and R. K. Trębiński. "Comparison of Computational Methods and Approaches Applied in Formulation of Boundary Conditions in Lagrange’s Ballistic Problem." Journal of Physics: Conference Series 2701, no. 1 (2024): 012132. http://dx.doi.org/10.1088/1742-6596/2701/1/012132.

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Abstract This paper compares algorithms for constructing a differential solution to the gas dynamics equations in the surrounding of a moving boundary. In order to compare the algorithms, the Lagrange problem, also known as the piston problem, was chosen as a test problem. A comparison was made between the author’s algorithm based on the method of characteristics and the classical approach with a fictitious cell using the solution of the Riemann problem. Calculations were carried out for the case of subsonic and supersonic gas motion. Comparisons were made on a fixed grid with a dynamically expanding cell (Euler grid) and on a uniformly expanding grid (Arbitrary Lagrangian-Eulerian – ALE grid). The problem was solved using numerical schemes of second order in time and space Lax - Wendroff type: the Richtmyer scheme and the MacCormack scheme. The results of calculations using the characteristics method were used as a benchmark solution. The comparative analysis carried out allows conclusions to be drawn regarding the accuracy of the individual approaches, as well as the scope of their applicability.

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Iwamoto, J. "Impingement of Under-Expanded Jets on a Flat Plate." Journal of Fluids Engineering 112, no. 2 (1990): 179–84. http://dx.doi.org/10.1115/1.2909385.

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When an under-expanded sonic jet impinges on a perpendicular flat plate, a shock wave forms just in front of the plate and some interesting phenomena can occur in the flow field between the shock and the plate. In this paper, experimental and numerical results on the flow pattern of this impinging jet are presented. In the experiments the flow field was visualized using shadow-photography and Mach-Zehnder interferometry. In the numerical calculations, the two-step Lax-Wendroff scheme was applied, assuming inviscid, axially symmetric flow. Some of the pressure distributions on the plate show that the maximum pressure does not occur at the center of the plate and that a region of reversed flow exists near the center of the plate.

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Kim, Hongjoong. "An efficient computational method for statistical moments of Burger's equation with random initial conditions." Mathematical Problems in Engineering 2006 (2006): 1–21. http://dx.doi.org/10.1155/mpe/2006/17406.

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The paper is concerned with efficient computation of numerical solutions to Burger's equation with random initial conditions. When the Lax-Wendroff scheme (LW) is expanded using the Wiener chaos expansion (WCE), random and deterministic effects can be separated and we obtain a system of deterministic equations with respect to Hermite-Fourier coefficients. One important property of the system is that all the statistical moments of the solution to the Burger's equation can be computed using the solution of the system only. Thus LW with WCE presents an alternative to computing moments by the Monte Carlo method (MC). It has been numerically demonstrated that LW with WCE approach is equally accurate but substantially faster than MC at least for certain classes of initial conditions.

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Hao, Tianchu, Yaming Chen, Lingyan Tang, and Songhe Song. "A third-order weighted nonlinear scheme for hyperbolic conservation laws with inverse Lax-Wendroff boundary treatment." Applied Mathematics and Computation 441 (March 2023): 127697. http://dx.doi.org/10.1016/j.amc.2022.127697.

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MUKHOPADHYAY, P. S., G. K. MANDAL, G. K. SEN, and D. K. SINHA. "A simple model to study wave-surge interaction." MAUSAM 48, no. 2 (2021): 323–28. http://dx.doi.org/10.54302/mausam.v48i2.4014.

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ABSTRACT. In this paper we have tried to set up a mathematical model that will show the contribution of wind-induced surface waves of the ocean, on surges in shallow basin of Bay of Bengal. For this, the energy balance equation, excluding non-linear forcing term, is considered and solved by Lax-Wendroff integration scheme. Wind is specified over all the grid points following Cardone' s formulation. The hydrodynamic equations in linearised form as used by Jelesnianski have been considered and using Shuman's algorithm, those equations have been solved. In the process of solving these equations, the output of the energy balance equation is included as wave set up term to incorporate energy contribution of wind waves to surges. The estimated surge height is compared with and without considering wave contribution.  

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Shermukhamedov, Abdulaziz, Nurillo Ergashev, and Abdukhamid Azizov. "Substantiating parameters brake system of the tractor trailer." E3S Web of Conferences 264 (2021): 04019. http://dx.doi.org/10.1051/e3sconf/202126404019.

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The article discusses substantiating the parameters brake system of a tractor-trailer (TT). The section offers a comparative analysis of theoretical and experimental studies of the TT brake drive and the parameters of its elements. Based on that, ordinary differential equations were solved by the Runge - Kutta method, the first-order accuracy (Euler's method). To solve partial differential equations, we used a modified Lax - Wendroff scheme. The results were obtained using the methods described above are theoretically very consistent with the triggering time ts = 0.47 s and the experimental value 0.46 s. Thus, the studying dynamic circuits of the pneumatic drive of TT brakes showed a high converging theoretical characteristic for a typical control line of a drive with an accelerating valve with experimental data, and the error was no more than 5%.

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Journal articles on the topic 'Lax-Wendroff scheme'

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