求解奇异退化扩散反应方程的高阶紧致差分格式及网格自适应方法

发布时间：2018-07-11 11:15

本文选题：奇异退化扩散反应方程 + 自适应方法　；参考：《宁夏大学》2016年硕士论文

【摘要】：研究奇异退化扩散反应方程解的爆破时间和爆破空间位置,具有重要的理论意义和实际应用价值.对此类方程的求解已经发展了许多数值算法.就目前的研究现状而言,主要是基于均匀网格上实施的,或者是基于低精度格式的自适应算法实施的,而关于奇异退化扩散反应方程的高精度紧致差分格式及网格自适应方法的研究非常少见.利用非均匀网格上的高精度紧致差分格式求解梯度变化较大的问题时,其计算精度相对于均匀网格的高精度紧致差分格式具有明显的优势.本文首先利用截断误差余项修正法建立了一维奇异退化扩散反应方程在非均匀网格上的高阶紧致差分格式,根据推导过程可知其时间具有二阶精度,空间具有三阶至四阶精度,网格是两层三点模版,可直接利用追赶法进行求解.在爆破点处温度关于时间的导数会产生大幅度的跳跃,本文利用等分布原理建立了时间和空间的网格自适应方法.由于网格产生了移动,所以利用线率对新旧网格上的函数值进行传递.并将方法从一维问题推广到高维问题.最后通过具有精确解的数值算例验证了本文方法的精确性和稳定性,再应用本文方法对爆破问题进行数值模拟,并与已有的数值结果进行比较,发现本文计算结果与文献中的数值结果相吻合.
[Abstract]:It is of great theoretical significance and practical application value to study the blasting time and space position of the solution of singular degenerate diffusion reaction equation. Many numerical algorithms have been developed for solving such equations. As far as the current research situation is concerned, it is mainly based on uniform mesh or adaptive algorithm based on low precision format. However, there are few studies on compact difference schemes and mesh adaptive methods for singular degenerate diffusion equations. When the high precision compact difference scheme on non-uniform mesh is used to solve the problem with large gradient variation, its computational accuracy is obviously superior to that of the high precision compact difference scheme of uniform grid. In this paper, first of all, by using the truncation error remainder correction method, the high order compact difference scheme of one-dimensional singular degenerate diffusion reaction equations on non-uniform grids is established. According to the derivation process, the time has second-order accuracy and the space has third-to fourth-order accuracy. The grid is a two-layer three-point template, which can be solved directly by the catch-up method. The derivative of temperature on time at the blasting point will produce a large jump. In this paper, the mesh adaptive method of time and space is established by using the principle of equal distribution. Because the grid is moving, the line rate is used to transfer the function values on the new and old grids. The method is extended from one-dimensional problem to high-dimensional problem. Finally, the accuracy and stability of the proposed method are verified by a numerical example with exact solution. The numerical simulation of the blasting problem is carried out by using the proposed method, and the numerical results are compared with the existing numerical results. It is found that the calculated results in this paper are in agreement with the numerical results in the literature.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

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