Study on the most suitable computational method for diffusion numerical simulation

1.考虑到二阶导数有限差分的高精度和稳定性，基于求解二阶波动方程代替一阶平流方程的思想，发展了一种新的平流精细格式。采用特征线法，得到了仅向下游传播的精确解。通过泰勒级数分析和大量的数值实验，可以确定该方法中涉及的参数是Courant数的函数。通过与其它各种格式的模型计算和VonNeumann稳定性分析的比较，证明了该格式优越的上级精度和稳定性。该方案通过分离特征曲线各分量方向，可方便地应用于多维实际问题。该格式只使用了三个计算网格点，因此不需要对边界进行过多的处理.为了准确有效地进行扩散数值模拟，对流项的计算格式和计算网格的大小都应引起重视。获得高精度结果的可用方案不仅取决于时间和空间上的网格间隔等计算条件，还取决于物理扩散和流速等水力条件，而如果选择用于数值模拟的方案，则存在最有效的网格尺寸以在允许的误差范围内获得精确解。我们试图发展一个标准来选择最有用的方案来计算平流项，并决定最有效的计算网格大小。我们用一个二阶数值扩散项来表示截断误差项，它是一个无穷级数。利用二阶数值扩散系数建立了该判据。通过一维试验扩散模拟验证了该判据的有效性

1. Taking into account the highly accurate and stable features of the finite difference of second order derivative, the new refined scheme for advection is developed based on the concept of solving 2nd order wave equation instead of 1st order advection equation. Characteristics method is used in order to get an accurate solution propagating downstream only. From Taylor series analysis and many numerical experiments, parameters involved in this method could be determined as functions of Courant number. Comparison of this scheme with the other various ones in model calculations and Von Neumann stability analysis prove its superior accuracy and stability. This scheme can easily be applied to multidimensional practical problems by separating characteristic curve each component direction. This proposed scheme uses only three computational grid points, so that there is no need to pay much attention to the treatment at the boundary.2. On making accurately and effectively a numerical diffusion simulation, one should pay much attention to both the computational scheme for calculating the advection term and the computational grid size. The usable schemes for obtaining the high-accurate results depend on not only computational conditions such as grid intervals on time and space but also hydraulic conditions such as physical diffusion and velocity, while there will be the most effective grid size to get accurate solution within the allowable margin of error if the scheme used for the numerical simulation is chosen. We have attempted to develop a criterion for selecting the most usable scheme to calculate the advection term and deciding the most effective computational grid size. We made a 2nd order numerical diffusion term represent the truncation error terms, which is a infinite series. The criterion was made up by utilizing the 2nd order numerical diffusivity. Some one-dimensional test diffusion simulations have been carried out to inspect the validity of the criterion