高分辨率有限差分方法及其应用研究
发布时间：2018-12-15 17:50
【摘要】：许多复杂的流动现象都可以通过计算流体力学的手段进行数值模拟。非线性双曲型守恒律方程作为流体运动的基本控制方程,对其数值方法的研究有着重要的科学意义和应用价值。本文主要研究双曲型守恒律方程的高分辨率有限差分方法,并将其应用于各种典型的数值算例,主要内容与研究结果包括以下几个方面:1.提出了一种基于WENO重构的熵稳定格式。以熵稳定数值通量为基础,通过在单元交界面处进行高阶WENO重构,得到一类高分辨率的数值格式,并用逐维计算的方法将其推广至二维情形。运用本格式对一维标量方程、一维气体动力学Euler方程组、一维浅水方程组和二维浅水方程组进行了大量的数值试验,并与原熵稳定格式的计算结果比较,结果表明基于WENO重构的熵稳定格式能有效提高解在间断处的分辨率。2.通过引入开关函数矩阵,提出了一种求解Euler方程组的自调节熵稳定格式。开关函数具有在光滑区域接近于0,而在间断区域接近于1的性质。该函数自动控制格式在不同位置的数值耗散大小,使得数值耗散在间断区域自动地添加,从而达到格式的自调节性。给出了一维和二维Euler方程组的几个经典算例,验证了该格式的良好性能。3.提出了一种求解双曲型守恒律方程的三阶熵稳定格式。首先基于不同模板上的两点熵守恒通量的线性组合得到四阶熵守恒通量;其次提出一种基于点值的满足符号性质的三阶基本无振荡重构,利用该重构进行(特征)熵变量重构,设计了一种三阶数值耗散项,将之添加到四阶熵守恒通量,得到了一种三阶熵稳定格式,并推广至二维情形。最后通过大量的数值试验来检验该格式的数值精度和有效性。数值结果表明,该格式在一维和二维情形下均能达到预期的三阶精度,在处理间断问题时具有高分辨率、基本无振荡性等优点。4.提出了一种求解双曲型守恒律方程的四阶半离散中心迎风格式。在Godunov型中心格式的基础上,充分考虑非线性波的局部传播速度,利用该速度对Riemann扇的宽度加以精确估计,得到了半离散中心迎风数值通量。将其与Peer的四阶基本无振荡重构相结合,建立了一种四阶半离散中心迎风格式。该格式无需求解Riemann问题,从而避免了复杂耗时的特征分解过程。运用该格式求解了标量守恒律方程、Euler方程组以及带坡底源项的浅水方程组。数值结果表明,该格式能准确地计算出解的复杂细小结构,具有高分辨率、基本无振荡、简单等优良特性。
[Abstract]:Many complex flow phenomena can be numerically simulated by means of computational fluid dynamics. As the basic governing equation of fluid motion, the nonlinear hyperbolic conservation law equation has important scientific significance and application value in the study of its numerical method. In this paper, the high resolution finite difference method for hyperbolic conservation law equations is studied and applied to various typical numerical examples. The main contents and results are as follows: 1. An entropy stable scheme based on WENO reconstruction is proposed. Based on the entropy stable numerical flux, a class of high resolution numerical schemes are obtained by reconstructing high order WENO at the interface of the unit. The scheme is extended to two dimensional cases by using the method of dimensionality calculation. By using this scheme, a large number of numerical experiments have been carried out on one-dimensional scalar equations, one-dimensional gas-dynamics Euler equations, one-dimensional shallow water equations and two-dimensional shallow-water equations, and the results are compared with those of the original entropy stability scheme. The results show that the entropy stable scheme based on WENO reconstruction can effectively improve the resolution at the discontinuity. 2. By introducing the switching function matrix, a self-adjusting entropy stabilization scheme for solving Euler equations is proposed. The switching function is close to 0 in the smooth region and close to 1 in the discontinuous region. This function automatically controls the numerical dissipation size of the scheme at different positions, which makes the numerical dissipation be added automatically in the discontinuous region, thus achieving the self-adjustment of the scheme. Several classical examples of one and two dimensional Euler equations are given, and the good performance of the scheme is verified. A third order entropy stable scheme for solving hyperbolic conservation law equations is proposed. Firstly, based on the linear combination of two-point entropy conservation flux on different templates, the fourth order entropy conservation flux is obtained. Secondly, a third order nonoscillatory reconstruction based on the point value satisfying the symbolic property is proposed. The entropy variable is reconstructed by this reconstruction, and a third order numerical dissipation term is designed, which is added to the fourth order entropy conservation flux. A third order entropy stable scheme is obtained and extended to two dimensional cases. Finally, the numerical accuracy and validity of the scheme are verified by a large number of numerical experiments. The numerical results show that the scheme can achieve the expected third-order accuracy in the case of one and two dimensions, and has the advantages of high resolution and basic non-oscillation in dealing with the discontinuous problems. 4. A fourth order semi-discrete central upwind scheme for solving hyperbolic conservation law equations is proposed. Based on the Godunov central scheme, the local propagation velocity of nonlinear wave is fully considered, and the width of Riemann fan is estimated accurately by using this velocity, and the numerical flux of semi-discrete center upwind is obtained. A fourth order semi-discrete central upwind scheme is established by combining it with the fourth order nonoscillatory reconstruction of Peer. The scheme does not need to solve the Riemann problem, thus avoiding the complex and time-consuming feature decomposition process. By using this scheme, the scalar conservation law equations, Euler equations and shallow water equations with source term at the slope bottom are solved. The numerical results show that the scheme can accurately calculate the complex and fine structure of the solution, and has the advantages of high resolution, basically no oscillation and simplicity.
【学位授予单位】：西北工业大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82
本文编号：2381053
[Abstract]:Many complex flow phenomena can be numerically simulated by means of computational fluid dynamics. As the basic governing equation of fluid motion, the nonlinear hyperbolic conservation law equation has important scientific significance and application value in the study of its numerical method. In this paper, the high resolution finite difference method for hyperbolic conservation law equations is studied and applied to various typical numerical examples. The main contents and results are as follows: 1. An entropy stable scheme based on WENO reconstruction is proposed. Based on the entropy stable numerical flux, a class of high resolution numerical schemes are obtained by reconstructing high order WENO at the interface of the unit. The scheme is extended to two dimensional cases by using the method of dimensionality calculation. By using this scheme, a large number of numerical experiments have been carried out on one-dimensional scalar equations, one-dimensional gas-dynamics Euler equations, one-dimensional shallow water equations and two-dimensional shallow-water equations, and the results are compared with those of the original entropy stability scheme. The results show that the entropy stable scheme based on WENO reconstruction can effectively improve the resolution at the discontinuity. 2. By introducing the switching function matrix, a self-adjusting entropy stabilization scheme for solving Euler equations is proposed. The switching function is close to 0 in the smooth region and close to 1 in the discontinuous region. This function automatically controls the numerical dissipation size of the scheme at different positions, which makes the numerical dissipation be added automatically in the discontinuous region, thus achieving the self-adjustment of the scheme. Several classical examples of one and two dimensional Euler equations are given, and the good performance of the scheme is verified. A third order entropy stable scheme for solving hyperbolic conservation law equations is proposed. Firstly, based on the linear combination of two-point entropy conservation flux on different templates, the fourth order entropy conservation flux is obtained. Secondly, a third order nonoscillatory reconstruction based on the point value satisfying the symbolic property is proposed. The entropy variable is reconstructed by this reconstruction, and a third order numerical dissipation term is designed, which is added to the fourth order entropy conservation flux. A third order entropy stable scheme is obtained and extended to two dimensional cases. Finally, the numerical accuracy and validity of the scheme are verified by a large number of numerical experiments. The numerical results show that the scheme can achieve the expected third-order accuracy in the case of one and two dimensions, and has the advantages of high resolution and basic non-oscillation in dealing with the discontinuous problems. 4. A fourth order semi-discrete central upwind scheme for solving hyperbolic conservation law equations is proposed. Based on the Godunov central scheme, the local propagation velocity of nonlinear wave is fully considered, and the width of Riemann fan is estimated accurately by using this velocity, and the numerical flux of semi-discrete center upwind is obtained. A fourth order semi-discrete central upwind scheme is established by combining it with the fourth order nonoscillatory reconstruction of Peer. The scheme does not need to solve the Riemann problem, thus avoiding the complex and time-consuming feature decomposition process. By using this scheme, the scalar conservation law equations, Euler equations and shallow water equations with source term at the slope bottom are solved. The numerical results show that the scheme can accurately calculate the complex and fine structure of the solution, and has the advantages of high resolution, basically no oscillation and simplicity.
【学位授予单位】：西北工业大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82
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