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Well-balanced central upwind schemes.

机译：平衡的中央迎风方案。

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摘要

Flux gradient terms and source terms are two fundamental components of hyperbolic systems of balance law. Though having distinct mathematical natures, they form and maintain an exact balance in a special class of solutions, which are called steady-state solutions. In this dissertation, we are interested in the construction of well-balanced schemes, which are the numerical methods for hyperbolic systems of balance laws that are capable of exactly preserving steady-state solutions on the discrete level.;We first introduce a well-balanced scheme for the Euler equations of gas dynamics with gravitation. The well-balanced property of the designed scheme hinges on a reconstruction process applied to equilibrium variables---the quantities that stay constant at steady states. In addition, the amount of numerical viscosity is reduced in the areas where the flow is in (near) steady-state regime, so that the numerical solutions under consideration can be evolved in a well-balanced manner.;We then consider the shallow water equations with friction terms, which become very stiff when the water height is close to zero. The stiffness in the friction terms introduces additional difficulty for designing an efficient well-balanced scheme. If treated explicitly, the stiff friction terms impose a severe restriction on the time step. On the other hand, a straightforward (semi-) implicit treatment of the stiff friction terms can greatly enhance the efficiency, but will break the well-balanced property of the resulting scheme. To this end, we develop a new semi-implicit Runge-Kutta time integration method that is capable of maintaining the well-balanced property under the time step restriction determined exclusively by non-stiff components in the underlying equations.;The well-balanced property of our schemes are tested and verified by extensive numerical simulations, and notably, the obtained numerical results clearly indicate that the well-balanced property plays an important role in achieving high resolutions when a coarse grid is used.
机译：通量梯度项和源项是双曲平衡律系统的两个基本组成部分。尽管它们具有独特的数学性质，但它们在一类特殊的解决方案（称为稳态解决方案）中形成并保持了精确的平衡。在本文中，我们对构造良好的平衡方案感兴趣，这是一种能够在离散级上精确地保持稳态解的双曲平衡律系统的数值方法。引力下的气体动力学欧拉方程的一种方案。设计方案的均衡特性取决于应用于均衡变量的重建过程，均衡变量是在稳态下保持不变的量。此外，在（近）稳态状态下流动的区域，数值粘度的量减少了，因此所考虑的数值解可以以平衡的方式演化。带有摩擦项的方程，当水位接近于零时变得非常僵硬。摩擦方面的刚度给设计有效的良好平衡方案带来了额外的困难。如果进行了明确处理，则刚性摩擦项会严重限制时间步长。另一方面，对刚性摩擦项进行直接（半）隐式处理可以大大提高效率，但会破坏所得方案的均衡特性。为此，我们开发了一种新的半隐式Runge-Kutta时间积分方法，该方法能够在仅由基础方程中的非刚性分量确定的时间步长限制下保持良好平衡的属性。我们的方案中的方案已通过广泛的数值模拟进行了测试和验证，值得注意的是，获得的数值结果清楚地表明，当使用粗糙网格时，良好的平衡特性对于实现高分辨率具有重要作用。

著录项

  • 作者

    Cui, Shumo.;

  • 作者单位

    Tulane University School of Science and Engineering.;

  • 授予单位 Tulane University School of Science and Engineering.;
  • 学科 Mathematics.;Applied mathematics.
  • 学位 Ph.D.
  • 年度 2015
  • 页码 131 p.
  • 总页数 131
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 物理化学（理论化学）、化学物理学;
  • 关键词

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