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双曲系统格式的正性原理。正性原理的基本原理是稳定性，这是数值格式的一个重要要求。正性原理是求解多维双曲系统方案的第一稳定性原理。在这个建议中，我们证明了Kurganov和Tadmor研究的中心方案是正的。将逆风方案与Lax-Wendroff方案相结合，得到了成本仅为原正方案30%的正方案。我们开发了一种称为凸本质非振荡（ENO）方案。凸ENO格式是一种高阶精确中心格式。我们开发了一种新的多重网格法来求解守恒双曲系统。通过采用多网格，大大降低了计算成本。在建议中，我们还提出了几种解决由界面分离的多流体椭圆问题的方案。这样的问题出现在许多实际应用程序中。例如，不可压缩多流体Navier-Stokes方程。为此，提出了一种新的非体拟合网格均匀二阶精确格式。本文提出了一种用有限元法求解具有混合边界条件的椭圆型问题的统一二阶精确水平集方法。这些问题是在利用孤岛动力学模型模拟外延薄膜生长过程中出现的。我们利用其中的一些方法对多相湍流进行了直接数值模拟。我们在鬼流法的基础上开发了求解此类椭圆型问题的几何多重网格方法，并计划对其他方法进行更多的研究。私家侦探和他的合作者正在寻求进一步发展积极的计划。在一系列的研究工作中，他们引入并建立了求解双曲守恒律系统的方案的正性原理。正性原则的基本原理是稳定性。1)他们证明了Kurganov和Tadmor开发的中心方案是正方案。2)他们继续开发一种新的积极方案，这是逆风和拉克斯-温德罗夫方案的混合。因此，不需要两阶段龙格-库塔，并且对于二维计算成本可以降低多达70%。3)他们继续研究一种新的方案，称为加权组件明智的积极方案。它是加权ENO方案和二阶分量的凸ENO方案或高分辨率中心方案的混合。他们使用所有候选点的凸组合来进行重建，但使用一种称为准确性的新测量方法而不是平滑度来分配适当的权重。凸组合几乎达到最优（比最优低一个阶）的顺序精度。该格式也可以推广到求解多维的Hamilton-Jacobi方程。他们将引入一种多重网格法来求解具有守恒定律的多维双曲系统。新颖之处在于在粗网格上计算通量，然后将通量的差值或通量的差值插值到细网格上。这种多网格方法在每次迭代中不仅比基本方案更快，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进一步推进了解决方案。这对于计算固定解特别有用。近年来，PI和他的合作者一直在寻求进一步发展鬼流体法（GFM）的多相流体。1)他们提出了一种几何多重网格方法来解决二维和三维中涉及多个界面的不规则边界问题引起的线性系统。在这种方法中，它们采用无矩阵的方法，即它们不显式地形成细网格矩阵，也不形成和存储粗网格矩阵。主要思想是构造一个精确的插值，通过水平集函数在接口处捕获正确的边界条件。2)提出了用有限元法求解具有Robin界面条件的椭圆方程的二阶精确水平集方法。他们首先研究了它的弱公式，然后证明存在一个唯一的弱解。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体拟合三角测量。3)提出了一种新的非体拟合网格二阶精确数值求解带界面椭圆方程的方法。而不是光滑的，边界和子域的边界以及接口，只需要作为子流形是利普希茨连续的。利用分段线性函数对弱公式进行离散，推导出弱公式的数值计算方法。当界面光滑或界面不连续点处理得当时，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)采用边界条件捕获法对多相湍流进行直接数值模拟。这是此类问题的首次成功DNS。因为湍流发生在很大的尺度范围内，因此需要非常有效的方法来捕获所有有意义的尺度。该建议侧重于现实世界的应用程序。例如，守恒定律的双曲系统，带界面的不可压缩Navier-Stokes方程，利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，是一种非常有效的方法。为了进一步提高数值模拟的速度，提出了两种多网格方法与这些方法相结合。该提案应具有广泛的影响，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等许多其他应用领域。