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.

在一系列研究工作中，我们介绍并建立了解决保护法的双曲线系统方案的积极性原则。 积极原则的基本原理是稳定性，这是数值方案非常重要的要求。 阳性原理是解决多维双曲系统方案的第一稳定性原理。 在此提案中，我们表明，库尔加诺夫和塔德莫尔研究的中心方案是正面的。 通过混合前风计划和Lax-Wendroff计划，我们制定了一个正面计划，该计划仅占原始正面计划的30％。我们已经开发了一种称为凸的方案，该方案本质上是非振荡（ENO）方案。 凸ENO方案是高阶准确中心方案。 我们已经开发了一种新的Multigrid方法来解决保护法的双曲线系统。通过进行多机，计算成本大大降低。在提案中，我们还开发了几个方案，以解决界面分开的多流体问题的椭圆问题。这些问题来自许多现实世界的应用。例如，不可压缩的多流体Navier-Stokes方程。为此开发了一个新的均匀的第二阶准确方案。我们已经使用有限元方法提出了统一的第二阶准确级别方法，以解决与边界条件混合条件的椭圆形问题。这些问题来自使用岛动力学模型模拟外延薄膜生长的出现。 我们已经使用了其中一些方法对多相湍流进行直接数值模拟。 我们已经开发了一种基于幽灵流体方法的椭圆形问题的几何多机方法，并计划使用其他方法做更多的事情。 PI及其合作者正在追求积极计划的进一步发展。在一系列研究中，他们介绍并确定了解决保护法的双曲线系统方案的积极原则。积极原则的理由是稳定。 1）他们证明了Kurganov和Tadmor开发的中心方案是积极的方案。 2）他们继续制定一个新的积极方案，该计划是前风和宽松的Wendroff计划的混合体。因此，不需要两个阶段的runge-kutta，对于二维，计算成本可以削减多达70％.3）。它是加权ENO方案的混合物和第二阶组件的凸面ENO方案或高分辨率中央方案的混合物。 他们使用所有候选人的凸组合来进行重建，但使用一种称为准确性的新测量，而不是平滑度来分配适当的权重。凸组合获得几乎最佳（低于最佳）订单精度。该方案也可以扩展以解决多维中的汉密尔顿 - 雅各比方程。 4）他们将引入一种用于解决多维夸张的保护法系统的多机方法。新颖的是计算粗网格上的通量，然后将通量或通量的差异插入到最好的网格上。这种多机方法不仅比每次迭代中的基本方案都要快，而且还允许比基本方案更大的时间步骤。因此，与基本方案相比，多机方法的CPU需要更少的CPU时间才能将解决方案提高到相同的停止时间。换句话说，在相同的CPU时间内，Multigrid方法在时间上进一步推进了解决方案。这对于计算固定解决方案特别有用。近年来，PI和他的合作者一直在为多相流体提供进一步的幽灵液法（GFM）的开发。 1）他们提出了一种几何多机方法来求解涉及2D和3D中多个接口的不规则边界问题引起的线性系统。在这种方法中，它们采用了无基质方法，即不会明确形成细网格矩阵，也永远不会形成或存储粗网格矩阵。主要思想是构建准确的插值，该插值通过级别集合函数捕获界面处的正确边界条件。 2）他们使用有限元方法提出了第二阶准确的级别方法，用于求解具有罗宾界面条件的椭圆方程。他们首先研究了它的弱制剂，然后证明存在独特的弱解决方案。最后，使用非体拟合均匀或任意三角剖分的有限元方法来解决这种弱的公式。这项工作的新颖性是有限元方法和非体拟合三角形的结合。 3）他们在非体拟合网格上开发了一种新的第二阶准确数值方法，用于求解使用接口的椭圆方程。边界和子域的束和界面，而不是光滑，只需要将Lipschitz连续为submanifold。开发出弱的公式，并通过通过零件线性函数离散弱制剂来得出数值方法。如果接口平滑或不连续处理，则该方法在最大规范中的第二阶准确是准确的，否则最大的规范会收敛。 4）他们使用捕获的边界条件捕获方法对多相湍流流进行直接数值模拟。这是此类问题的第一个成功的DN。由于湍流是通过大量尺度进行的，因此需要非常有效的方法来捕获所有有意义的量表。该建议集中于现实世界应用。例如，保护定律的倍重系统，不可压缩的Navier-Stokes方程，具有界面，使用岛动力学模型的外延薄膜生长，多相湍流流的直接数值模拟。所提出的数值方法具有高阶精度和高分辨率，因此它们非常有效。 提出了两种多机方法与这些方法相结合，以进一步加快数值模拟。该提案应该具有广泛的影响，因为创建的方法可以轻松地用于环境，地球物理，生物学，材料科学和工程科学中的许多其他应用领域。