温度场影响下浅水流模型的有限差分WENO格式

发布时间：2018-03-15 22:35

本文选题：Ripa模型　切入点：有限差分WENO格式　出处：《青岛大学》2017年硕士论文　论文类型：学位论文

【摘要】：Ripa模型是在浅水波方程的基础上增加了水面温度梯度,由于水体底部不平而带有几何源项,属于双曲平衡律方程。值得注意的是Ripa模型保持定常解,即流通量梯度非零且与源项保持精确平衡。针对源项的标准的数值处理无法保持定常解,容易导致伪振荡,即使采用网格加细策略也无法消除振荡。能够在离散状态下保持定常解的数值格式被称作well-balanced格式。本论文中,针对Ripa模型我们通过将源项进行特殊分裂以及设计高阶线性有限差分算子来逼近流通量梯度以及源项中的导数,建立高阶well-balanced有限差分WENO格式。严格的理论分析以及广泛的数值结果都表明该格式保持well-balanced性质。此外,一维、二维数值结果还显示格式具有高阶精度,同时针对间断解拥有高分辨率。
[Abstract]:The Ripa model is a hyperbolic equilibrium law equation, which increases the temperature gradient of the water surface on the basis of the shallow water wave equation. Due to the uneven bottom of the water body, it belongs to the hyperbolic equilibrium law equation. It is worth noting that the Ripa model maintains a steady solution. That is, the flux gradient is non-zero and keeps an accurate balance with the source term. The standard numerical treatment of the source term can not maintain a steady solution, which can easily lead to pseudo oscillation. Even the mesh refinement strategy can not eliminate the oscillation. The numerical scheme, which can keep the steady solution in discrete state, is called well-balanced scheme. For the Ripa model, we approach the flux gradient and the derivative of the source term by splitting the source term and designing a higher-order linear finite-difference operator. The high order well-balanced finite difference WENO scheme is established. Strict theoretical analysis and extensive numerical results show that the scheme preserves the well-balanced property. In addition, one and two dimensional numerical results also show that the scheme has higher order accuracy. At the same time, the discontinuous solution has high resolution.
【学位授予单位】：青岛大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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