浅水方程大时间步长格式源项的处理方法

Numerical simulation of water is one of the most important part of researching fluvial processes. Study of the Numerical Scheme can improve the computational efficiency, so it has realistic significance. Numerical Scheme has always been a hot spot in international as a basic theory research of mechanics and new schemes emerged from day to day. The applicant has built the new scheme (Large Time Step Scheme) which enhanced the efficiency about 10 times more than traditional schemes with assurance of high accuracy in doctoral period. This research is the extension of the work in doctoral period, and in this research the source terms of the equations need to be handled in the scheme. The Large Time Step scheme of shallow flows based on Exact Riemann Solver had been presented but without considering source terms. The numerical scheme is not balanced because the false flow will be introduced in when the source terms are discretized separately. The balance of numerical scheme had been studied well by formers and kinds of well balanced traditional schemes were proposed which can be referenced to extend to Large Time Step scheme. However, some difference still exists because the simple wave with source term will go through more than one element in one time step. It is necessary to work out how the simple waves with source terms interact when going through elements to make sure the well balanced Large Time Step scheme of shallow water flows could be established.

水流数值模拟是研究河流演变的重要组成部分。数值模拟的格式是力学的基础理论研究，通过对数值模拟格式的研究，计算速度不断得到提高，因此具有十分重要的现实意义，也是国际研究的热点之一，新的数值格式层出不穷。申请者在博士阶段已经建立了不含源项浅水方程大时间步长格式，在保证计算精度的同时将计算效率提高了十倍以上。本研究是博士阶段研究的进一步扩展和深入，主要解决大时间步长格式的源项处理方法。现有研究已经实现了基于黎曼精确解的大时间步长格式，但是没有考虑源项。源项与对流项分开考虑使得数值格式在水体处于平衡状态时产生虚假流动，这就是所谓的和谐性问题。传统数值的格式的和谐性问题已经得到了充分的研究并已提出了很多相应的解决方法，可以作为本研究的参考。但是传统格式的和谐性又与大时间步长格式下的和谐性有着本质的区别。在大时间步长格式中，单波会在一个时间步长内穿过多个单元，那么携带源项的单波与所穿过单元如何相互作用的，这是本研究需要着力解决的问题。

本研究的前期工作实现了齐次浅水方程的大时间步长格式，数学实验模型是溃坝，考虑底坡溃坝问题称为阶梯黎曼问题，但是这已经不能检验大时间步长格式的广泛实用性，因为阶梯黎曼问题只在间断部分有一个阶梯，整个计算域只有一个界面符合条件，不具有普适性。因此选取和谐格式算例flow over a hump，这个算例离散后具有多个阶梯黎曼问题，有正坡有负坡。通过三年研究，实现了近似黎曼解的大时间步长格式，最大CFL数可以达到5。但是震荡无法消除，虽然传统格式有很多抑制震荡的方法，但是大都不适用于大时间步长格式，目前为止，提出抑制震荡的方法极少，因此这个领域是待研究的。另外还研究了阶梯黎曼问题的精确解，对于正坡的情形，目前还没有得到解决，因此无法将其应用于大时间步长格式。