• 主题
高级搜索  >
历史搜索 清空历史
首页> 外文学位 >A higher-order conservation element solution element method for solving hyperbolic differential equations on unstructured meshes.
【24h】

A higher-order conservation element solution element method for solving hyperbolic differential equations on unstructured meshes.

机译：非结构网格上双曲型微分方程的高阶守恒元素解单元法。

获取原文
获取原文并翻译 | 示例
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

This dissertation presents an extension of the Conservation Element Solution Element (CESE) method from second- to higher-order accuracy. The new method retains the favorable characteristics of the original second-order CESE scheme, including (i) the use of the space-time integral equation for conservation laws, (ii) a compact mesh stencil, (iii) the scheme will remain stable up to a CFL number of unity, (iv) a fully explicit, time-marching integration scheme, (v) true multidimensionality without using directional splitting, and (vi) the ability to handle two- and three-dimensional geometries by using unstructured meshes. This algorithm has been thoroughly tested in one, two and three spatial dimensions and has been shown to obtain the desired order of accuracy for solving both linear and non-linear hyperbolic partial differential equations. The scheme has also shown its ability to accurately resolve discontinuities in the solutions.;Higher order unstructured methods such as the Discontinuous Galerkin (DG) method and the Spectral Volume (SV) methods have been developed for one-, two- and three-dimensional application. Although these schemes have seen extensive development and use, certain drawbacks of these methods have been well documented. For example, the explicit versions of these two methods have very stringent stability criteria. This stability criteria requires that the time step be reduced as the order of the solver increases, for a given simulation on a given mesh.;The research presented in this dissertation builds upon the work of Chang, who developed a fourth-order CESE scheme to solve a scalar one-dimensional hyperbolic partial differential equation. The completed research has resulted in two key deliverables. The first is a detailed derivation of a high-order CESE methods on unstructured meshes for solving the conservation laws in two- and three-dimensional spaces. The second is the code implementation of these numerical methods in a computer code. For code development, a one-dimensional solver for the Euler equations was developed. This work is an extension of Chang's work on the fourth-order CESE method for solving a one-dimensional scalar convection equation. A generic formulation for the nth-order CESE method, where n ≥ 4, was derived. Indeed, numerical implementation of the scheme confirmed that the order of convergence was consistent with the order of the scheme. For the two- and three-dimensional solvers, SOLVCON was used as the basic framework for code implementation. A new solver kernel for the fourth-order CESE method has been developed and integrated into the framework provided by SOLVCON. The main part of SOLVCON, which deals with unstructured meshes and parallel computing, remains intact. The SOLVCON code for data transmission between computer nodes for High Performance Computing (HPC).;To validate and verify the newly developed high-order CESE algorithms, several one-, two- and three-dimensional simulations where conducted. For the arbitrary order, one-dimensional, CESE solver, three sets of governing equations were selected for simulation: (i) the linear convection equation, (ii) the linear acoustic equations, (iii) the nonlinear Euler equations. All three systems of equations were used to verify the order of convergence through mesh refinement. In addition the Euler equations were used to solve the Shu-Osher and Blastwave problems. These two simulations demonstrated that the new high-order CESE methods can accurately resolve discontinuities in the flow field.For the two-dimensional, fourth-order CESE solver, the Euler equation was employed in four different test cases. The first case was used to verify the order of convergence through mesh refinement. The next three cases demonstrated the ability of the new solver to accurately resolve discontinuities in the flows. This was demonstrated through: (i) the interaction between acoustic waves and an entropy pulse, (ii) supersonic flow over a circular blunt body, (iii) supersonic flow over a guttered wedge. To validate and verify the three-dimensional, fourth-order CESE solver, two different simulations where selected. The first used the linear convection equations to demonstrate fourth-order convergence. The second used the Euler equations to simulate supersonic flow over a spherical body to demonstrate the scheme's ability to accurately resolve shocks. All test cases used are well known benchmark problems and as such, there are multiple sources available to validate the numerical results. Furthermore, the simulations showed that the high-order CESE solver was stable at a CFL number near unity.
机译：本文提出了守恒元素解元方法从二阶精度到高阶精度的扩展。新方法保留了原始二阶CESE方案的有利特征，包括（i）使用时空积分方程守恒定律，（ii）紧凑的网格模具，（iii）该方案将保持稳定达到CFL的单位数；（iv）完全明确的时间行进整合方案；（v）真正的多维性，而无需使用方向分割；（vi）通过使用非结构化网格处理二维和三维几何的能力。该算法已在一个，两个和三个空间维度上进行了全面测试，并已显示出获得线性和非线性双曲型偏微分方程求解所需的精度顺序。该方案还显示了其能够准确解决溶液中不连续性的能力。高阶非结构化方法，例如不连续Galerkin（DG）方法和光谱体积（SV）方法，已针对一维，二维和三维方法进行了开发应用。尽管这些方案已得到广泛的开发和使用，但这些方法的某些缺点已得到充分证明。例如，这两种方法的显式版本具有非常严格的稳定性标准。对于给定的网格上的给定模拟，此稳定性标准要求随着求解器阶数的增加，时间步长应减小。;本论文中的研究基于Chang的工作，他开发了四阶CESE方案来求解标量一维双曲偏微分方程。完成的研究得出了两个关键的可交付成果。首先是对非结构网格上高阶CESE方法的详细推导，以解决二维和三维空间中的守恒律。第二个是这些数字方法在计算机代码中的代码实现。为了进行代码开发，开发了用于Euler方程的一维求解器。这项工作是Chang所研究的用于解决一维标量对流方程的四阶CESE方法的扩展。推导了n阶CESE方法的一般公式，其中n≥4。实际上，该方案的数值实现证实了收敛的顺序与该方案的顺序是一致的。对于二维和三维求解器，SOLVCON被用作代码实现的基本框架。已开发出一种用于四阶CESE方法的新求解器内核，并将其集成到SOLVCON提供的框架中。 SOLVCON的主要部分保持不变，该部分处理非结构化网格和并行计算。用于高性能计算机（HPC）的计算机节点之间的数据传输的SOLVCON代码。为了验证和验证新开发的高阶CESE算法，进行了一些一维，二维和三维仿真。对于任意阶的一维CESE求解器，选择了三组控制方程进行仿真：（i）线性对流方程，（ii）线性声学方程，（iii）非线性Euler方程。这三个方程组均用于通过网格细化验证收敛的顺序。另外，使用欧拉方程来求解Shu-Osher和Blastwave问题。这两个模拟结果表明，新的高阶CESE方法可以准确地解决流场中的不连续性。对于二维四阶CESE求解器，在四个不同的测试案例中采用了Euler方程。第一种情况用于通过网格细化验证收敛的顺序。接下来的三个案例证明了新求解器能够准确解决流中的不连续性。这可以通过以下方式得到证明：（i）声波与熵脉冲之间的相互作用；（ii）圆形钝体上的超音速流；（iii）楔形沟槽上的超音速流。要验证和验证三维四阶CESE求解器，请选择两个不同的仿真。第一个使用线性对流方程来证明四阶收敛。第二种方法使用Euler方程来模拟超音速在球体上的流动，以证明该方案能够准确解决冲击。使用的所有测试用例都是众所周知的基准测试问题，因此，可以使用多种来源来验证数值结果。此外，仿真表明，高阶CESE求解器在CFL数接近于1时是稳定的。

著录项

  • 作者

    Bilyeu, David.;

  • 作者单位

    The Ohio State University.;

  • 授予单位 The Ohio State University.;
  • 学科 Mechanical engineering.;Aerospace engineering.;Applied mathematics.
  • 学位 Ph.D.
  • 年度 2014
  • 页码 229 p.
  • 总页数 229
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号