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1.
We introduce a new fourth order, semi-discrete, central-upwind scheme for solving systems of hyperbolic conservation laws. The scheme is a combination of a fourth order non-oscillatory reconstruction, a semi-discrete central-upwind numerical flux and the third order TVD Runge-Kutta method. Numerical results suggest that the new scheme achieves a uniformly high order accuracy for smooth solutions and produces non-oscillatory profiles for discontinuities. This is especially so for long time evolution problems. The scheme combines the simplicity of the central schemes and accuracy of the upwind schemes. The advantages of the new scheme will be fully realized when solving various examples. 相似文献
2.
给出了求解一维双曲型守恒律的一种半离散三阶中心迎风格式,并利用逐维进行计算的方法将格式推广到二维守恒律。构造格式时利用了波传播的单侧局部速度,三阶重构方法的引入保证了格式的精度。时间方向的离散采用三阶TVD Runge—Kutta方法。本文格式保持了中心差分格式简单的优点,即不需用Riemann解算器,避免了进行特征分解过程。用该格式对一维和二维守恒律进行了大量的数值试验,结果表明本文格式是高精度、高分辨率的。 相似文献
3.
Application of a fourth-order relaxation scheme to hyperbolic systems of conservation laws 总被引:5,自引:0,他引:5
A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions.
The scheme is based on a fourth-order central weighted essentially nonoscillatory (CWENO) reconstruction for one-dimensional
cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type
solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation
of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties
of the present method are demonstrated in one- and two-dimensional numerical experiments.
The project supported by the National Natural Science Foundation of China (60134010)
The English text was polished by Yunming Chen. 相似文献
4.
An approximate‐state Riemann solver for the solution of hyperbolic systems of conservation laws with source terms is proposed. The formulation is developed under the assumption that the solution is made of rarefaction waves. The solution is determined using the Riemann invariants expressed as functions of the components of the flux vector. This allows the flux vector to be computed directly at the interfaces between the computational cells. The contribution of the source term is taken into account in the governing equations for the Riemann invariants. An application to the water hammer equations and the shallow water equations shows that an appropriate expression of the pressure force at the interface allows the balance with the source terms to be preserved, thus ensuring consistency with the equations to be solved as well as a correct computation of steady‐state flow configurations. Owing to the particular structure of the variable and flux vectors, the expressions of the fluxes are shown to coincide partly with those given by the HLL/HLLC solver. Computational examples show that the approximate‐state solver yields more accurate solutions than the HLL solver in the presence of discontinuous solutions and arbitrary geometries. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
5.
Mohammed M. Babatin 《International Journal of Computational Fluid Dynamics》2013,27(10):723-735
In this article, we develop an adaptive scheme for solving systems of hyperbolic conservation laws. In this scheme nonlinear shock and linear contact waves will be treated differently. The proposed scheme uses the Kurganov central-upwind scheme. Fourth-order non-oscillatory reconstruction is employed near shock only while the unlimited fifth-order reconstruction is used for smooth regions and linear contact waves. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multi-resolution technique. The advantages of the scheme are high resolution and computationally efficient since limiters are used only for shocks. Numerical experiments with one- and two-dimensional problems are presented which show the robustness of the proposed scheme. 相似文献
6.
In this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ? ? [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
7.
In this work we present an upwind‐based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non‐linear problems by using the framework of relaxation system that converts a non‐linear conservation law into a system of linear convection equations with a non‐linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non‐standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
8.
孙鸿烈 《应用数学和力学(英文版)》1999,20(7):789-794
IntroductionFrompracticalproblem,wecouldconcludemanyproblemsaboutsolvingparabolicpartialdiferentialequation.Nowtherearemanynu... 相似文献
9.
Yousef Hashem Zahran 《国际流体数值方法杂志》2008,57(6):745-760
We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
10.
11.
一族新的高精度显式差分格式 总被引:1,自引:1,他引:0
对求解三维势物型偏微分方程,利用待定参数法构造出一族新的高精度的三层显式差分格式,其精度为O(△t^3+△x^4+△y^4+△z^4),并论证了其稳定性,通过数值实例可见其精度较文「1」提高2位有效数字。 相似文献
12.
13.
Christophe Berthon 《Flow, Turbulence and Combustion》2006,76(4):343-351
The numerical simulations of compressible flows need more and more accuracy. Several approaches can be used to increase the order of the scheme. One of the most popular has been introduced by [11]: the so-called MUSCL scheme. In the present work, we focus our attention on the robustness of this method. We propose a relevant CFL restriction and a new gradient reconstruction strategy to enforce stability of the method. The novelty stays in the fact that non conservation argument is involved in the gradient reconstruction. Numerical results are performed using this new variant of the MUSCL scheme. 相似文献
14.
In this paper, by the method of collocation, using the cubic β-spline function as the trial function in the time domain and
putting zero residuals of the differential equation of motion of the structure at two points of time, the authors obtain an
unconditionally stable calculation scheme for the dynamic response of the structure. When a parameter σ in the scheme is within
the interval 0.15<σ <0.5 the scheme is absolutely stable. It is shown that the accuracy of the scheme, as may be measured
by AD (the decay of the amplitudes), PE (the elongation of periods) and the algorithmic damping ratio, is better than that
of traditional methods—the Wilson-σ's method, the Newmark's method and the Houbolt's method. A numerical example is given
in which a certain dynamic response problem is solved by the method of this paper and results are compared with that of the
traditional methods and the analytic method showing that the accuracy of the method by this paper is superior to the other
ones. The computational scheme for the dynamic response of structures by this paper may be regarded as an effective, convenient
and accurate method for dynamic response of structures. 相似文献
15.
In this article, we present an improved third-order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO-ZQ scheme. However, the global smoothness indicator has a little different from WENO-ZQ scheme. In this follow-up article, a convex combination of a second-degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three-point information is adopted by the improved third-order WENO scheme, the truncation errors are smaller than some other third-order WENO schemes in L∞ and L2 norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one- and two-dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third-order WENO schemes. 相似文献
16.
刘继军 《应用数学和力学(英文版)》2003,24(5):605-613
A 3-layered explicit difference scheme for the numerical solution of 2-D heat equation is proposed. Firstly, a general symmetric
difference scheme is constructed and its optimal error is obtained. Then two kinds of condition for choosing the parameters
for optimal error and stable difference scheme are given. Finally, some numerical results are presented to show the advantage
of the schemes
Foundation items: the Science Foundation of Chinese Postdoctoral (2002031224); the Science Foundations of Southeast University (9209011148,
3007011043)
Biography: Liu Ji-jun (1965-) 相似文献
17.
A group of asymmetric difference schemes to approach the Korteweg-de Vries(KdV)equation is given here.According to such schemes,the full explicit difference scheme and the fun implicit one,an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed.The scheme is linear unconditionally stable by the analysis of linearization procedure,and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy. 相似文献
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19.
应用判别差分方程稳定性的Schur-Cohn准则,研究用于一般耦合系统动力响应分析的单步Newmark预测-校正算法的稳定性问题;给出了算法无条件稳定的充分必要条件的严格理论证明。 相似文献
20.
Eric Ngondiep 《国际流体数值方法杂志》2020,92(4):266-284
This work deals with the numerical solutions of two-dimensional viscous coupled Burgers' equations with appropriate initial and boundary conditions using a three-level explicit time-split MacCormack approach. In this technique, the differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second-order accurate in time and fourth-order convergent in space, whereas it is second-order convergent in both time and space for high Reynolds numbers problems. This observation shows the utility and efficiency of the considered method compared with a broad range of numerical schemes widely studied in the literature for solving the two-dimensional time-dependent nonlinear coupled Burgers' equations. A large set of numerical examples that confirm the theoretical results are presented and critically discussed. 相似文献