High Order and Efficient Numerical Schemes for Multi-Dimensional Hyperbolic Systems of Conservation Laws and for Simulations of Multi-Phase Fluids in Applications

守恒定律多维双曲系统和应用中多相流体模拟的高阶高效数值方案

• 批准号: 0411504
• 负责人: Carlos Garcia-Cervera
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-15 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0411504&HistoricalAwards=false
• 关键词: Order; Efficient; Numerical; Schemes; Multi

In a series of research works we have introduced and established thepositivity principle for schemes for solving hyperbolic systems of conservation laws. The rationale of the positivity principle is stability, which is a very important requirement for numerical schemes. The positivity principle is the 1st stability principle for schemes for solving multi-dimensional hyperbolic systems. In this proposal we have shown that the central scheme studied by Kurganov and Tadmor is positive. By mixing upwind scheme and Lax-Wendroff scheme, we have made a positive scheme which costs only 30% of the original positive scheme. We have developed a scheme called Convex Essentially Non-Oscillatory (ENO) scheme. The Convex ENO scheme is a high order accurate central scheme. We have developed a new multigrid method to solve hyperbolic systems of conservation laws. By doing multigrid, the cost of calculations is reduced significantly. In the proposal we also develop several schemes for solving elliptic problems with multi-fluids separated by the interfaces. Such problems arise from many real world applications. For example, incompressible multi-fluids Navier-Stokes equation. A new uniform 2nd order accurate scheme on non-body-fitting grids is developed for that. We have proposed a uniform 2nd order accurate level-set method using finite element method for solving elliptic problems with mixing boundary conditions. Such problems emerge from in simulating epitaxial thin film growth using the island dynamics model. We have used some of those methods to do Direct Numerical Simulation on multi-phase turbulent flows. We have developed a geometric multigrid method for such elliptic problems based on the Ghost Fluid Method, and plan to do more with the other methods. The PI and his collaborators are pursuing further development of positive schemes. In a series of research works they have introduced and established the positivity principle for schemes for solving hyperbolic systems of conservation laws. The rationale of the positivity principle is stability. 1) They prove that the central scheme developed by Kurganov and Tadmor is positive scheme. 2) They continue to develop a new positive scheme, which is a mixture of upwind and Lax-Wendroff schemes. Hence two-stage Runge-Kutta is not required and for two-dimensions the computation cost could be cut by as much as 70%.3) They continue to work on a new scheme called weighted component-wise positivescheme. It is a mixture of Weighted ENO schemes and 2nd order component-wise version of Convex ENO scheme or high-resolution central scheme. They use a convex combination of all candidates to do reconstruction, but use a new measurement called accurateness instead of smoothness to assign proper weights. The convex combination achieves almost optimal (one order lower than the optimal) order accuracy. This scheme can be also extended to solve Hamilton-Jacobi equations in multi-dimensions. 4) They are going to introduce a multigrid method for solving multi-dimensional hyperbolic systems of conservation laws. The novelty is to calculate the fluxes on coarse grid, then interpolates the differences of the fluxes or the fluxes to the finest grid. Such multigrid method is not only faster than a base scheme in each iteration, but also allows larger time step than that of the base scheme. Hence the multigrid method requires much less CPU time to advance solutions to the same stopping time compared to the base scheme. In other words, for the same CPU time, the multigrid method advances solutions much further in time. This is particularly useful for computing stationary solutions. In the recent years, the PI and his collaborators have been pursuing further development of Ghost Fluid Method (GFM) for multi-phase fluids. 1) They propose a geometric multigrid method to solving linear systems arising from irregular boundary problems involving multiple interfaces in 2D and 3D. In this method, they adopt a matrix-free approach i.e. they do not form the fine grid matrix explicitly and they never form nor store the coarse grid matrices. The main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function. 2) They propose a 2nd order accurate level-set method using finite element method for solving elliptic equations with Robin interface conditions. They first study a weak formulation of it, and then prove thatthere exists a unique weak solution. At last, a finite element method on non-body-fitting uniform or arbitrary triangulations is used to solve such weak formulation. The novelty of this work is the incorporation of finite element methods and non-body-fitting triangulations. 3) They develop a new 2nd order accurate numerical method on non-body-fitting grids for solving the elliptic equations with interfaces. Instead of smooth, the boundary and the subdomains'boundaries and hence the interfaces, are only required to be Lipschitz continuous as submanifold. A weak formulation is developed and the numerical method is derived by discretizing the weak formulation by piece-wise linear functions. The method is 2nd order accurate in maximum norm if the interface is smooth or its discontinuities are proper handled, and convergent in maximum norm otherwise. 4) They use the boundary condition capturing methodto do Direct Numerical Simulations on multi-phase turbulentflows. This is the first successful DNS of such problems.Because turbulence happens through a large range of scales, and hence very efficient methods are needed to capture all meaningful scales.The proposal focus on the real world applications. For example, hyperbolicsystems of conservation laws, incompressible Navier-Stokes equations with interfaces, epitaxial thin film growth using the island dynamics model, Direct Numerical Simulation on multi-phase turbulent flows. The proposed numerical methods possess high order accuracy and high resolutions, hence they are very efficient. Two multigrid methods are proposed to couple with those methods to further speeded up numerical simulations. The proposal should have broad impact, since the methods created can be easily adopted to many other application areas in the environmental, geophysical, biological, material science, and engineering sciences.

在一系列的研究工作中，我们引入并建立了求解守恒定律双曲系统的方案的实证性原理。 实证性原理的基本原理是稳定性，这是数值格式的一个非常重要的要求。 正性原理是解决多维双曲系统方案的第一稳定性原理。 在这个提案中，我们已经证明库尔加诺夫和塔德莫尔研究的中心方案是积极的。 通过混合迎风方案和Lax-Wendroff方案，我们得到了正方案，其成本仅为原正方案的30%。我们开发了一种称为凸本质非振荡（ENO）方案的方案。 凸ENO方案是一种高阶精确中心方案。 我们开发了一种新的多重网格方法来求解守恒定律的双曲系统。通过使用多重网格，计算成本显着降低。在该提案中，我们还开发了几种解决由界面分隔的多流体椭圆问题的方案。此类问题来自许多现实世界的应用。例如，不可压缩多流体纳维-斯托克斯方程。为此开发了一种新的非身体贴合网格的统一二阶精确方案。我们提出了一种使用有限元方法求解具有混合边界条件的椭圆问题的统一二阶精确水平集方法。这些问题是在使用岛动力学模型模拟外延薄膜生长时出现的。 我们已经使用其中一些方法对多相湍流进行直接数值模拟。 我们基于幽灵流体法开发了一种用于解决此类椭圆问题的几何多重网格方法，并计划使用其他方法做更多的事情。 PI 和他的合作者正在寻求积极计划的进一步发展。在一系列研究工作中，他们引入并建立了求解守恒定律双曲系统方案的实证性原理。积极性原则的基本原理是稳定性。 1）证明Kurganov和Tadmor提出的中心方案是正方案。 2) 他们继续开发一种新的正方案，该方案是迎风方案和 Lax-Wendroff 方案的混合体。因此，不需要两阶段龙格-库塔，并且对于二维，计算成本可以减少多达 70%。3) 他们继续研究一种称为加权分量正向方案的新方案。它是加权 ENO 方案和凸 ENO 方案或高分辨率中心方案的二阶分量版本的混合。 他们使用所有候选者的凸组合来进行重建，但使用称为准确度的新测量而不是平滑度来分配适当的权重。凸组合实现了几乎最优（比最优低一阶）的阶次精度。该方案还可以扩展到求解多维的 Hamilton-Jacobi 方程。 4）他们将引入多重网格方法来求解守恒定律的多维双曲系统。新颖之处在于计算粗网格上的通量，然后将通量的差值或细网格的通量插值。这种多重网格方法不仅在每次迭代中比基本方案更快，而且允许比基本方案更大的时间步长。因此，与基本方案相比，多重网格方法需要更少的 CPU 时间来将解决方案推进到相同的停止时间。换句话说，对于相同的 CPU 时间，多重网格方法在时间上将解决方案推进得更远。这对于计算平稳解特别有用。近年来，PI 和他的合作者一直致力于进一步开发用于多相流体的幽灵流体法（GFM）。 1) 他们提出了一种几何多重网格方法来解决涉及 2D 和 3D 多个界面的不规则边界问题所产生的线性系统。在这种方法中，它们采用无矩阵方法，即它们不显式地形成细网格矩阵，并且它们从不形成也不存储粗网格矩阵。主要思想是构造一个精确的插值，通过水平集函数捕获界面处的正确边界条件。 2）他们提出了一种使用有限元方法求解具有 Robin 接口条件的椭圆方程的二阶精确水平集方法。他们首先研究它的弱表述，然后证明存在唯一的弱解。最后，采用非体拟合均匀或任意三角剖分的有限元方法来求解这种弱公式。这项工作的新颖之处在于结合了有限元方法和非人体拟合三角测量。 3）他们开发了一种新的非身体拟合网格上的二阶精确数值方法，用于求解具有界面的椭圆方程。边界和子域的边界以及界面不需要平滑，只需要像子流形一样是 Lipschitz 连续的。开发了弱公式，并通过分段线性函数离散弱公式来导出数值方法。如果界面光滑或者其不连续性得到适当处理，则该方法在最大范数上是二阶精确的，否则在最大范数上是收敛的。 4）他们使用边界条件捕获方法对多相湍流进行直接数值模拟。这是此类问题的第一个成功的 DNS。因为湍流发生在大范围的尺度上，因此需要非常有效的方法来捕获所有有意义的尺度。该提案侧重于现实世界的应用。例如，守恒定律的双曲系统、具有界面的不可压缩纳维-斯托克斯方程、使用岛动力学模型的外延薄膜生长、多相湍流的直接数值模拟。所提出的数值方法具有高阶精度和高分辨率，因此非常有效。 提出了两种多重网格方法来与这些方法结合以进一步加速数值模拟。该提案应该具有广泛的影响，因为所创建的方法可以很容易地应用于环境、地球物理、生物、材料科学和工程科学的许多其他应用领域。