Numerical Amalysis and Glolal Behavioz of Chemotactic Equations

1998年，我们提出了一种基于有限元法和龙格-库塔法的化学趋化方程离散化方案，并从理论上证明了该方案的稳定性和近似解的收敛性。在证明中，引入了离散半群和离散演化算子的概念来精确地描述近似解。1999年，我们根据设计的方案建立了计算算法。如果对某些有限元方法使用常规算法，则需要大量的机器内存。所以在这项研究中，我们做了一些装置，我们用一种合适的方式交换矩阵的成分，以压缩矩阵带的大小。这样，空间变量在一维情况下可分为8192，在二维情况下可分为256。利用该算法对趋化方程进行了数值计算。作为结果，主要阐明了以下两点关于解的全局行为。随着方程中敏感函数的形式的加入，从溶液中得到的胞状霉菌的模式发生了很大的变化。对于具有生长项的趋化方程，在方程的解中确实观察到所需的模型类型，即同心圆和分支。

In 1998 we devised a discretization scheme for the chemotactic equations which is based on the finite element methods and the Runge-Kutta methods, and proved theoretically stability of the scheme and convergence of the approximate solution. In the proof, notions of the discrete semigroup and discrete evolution operator were newly introduced to describe the approximate solutions precisely.In 1999 we set up algorithm for calculations by the scheme devised. If one uses usual algorithm for some finite element method, enormous memories of machine are needed. So in this research we made some device that we exchange components of the matrix in a suitable way in order to condense a size of the band of the matrix. By this the spatial variable can be divided into 8192 in the one dimensional case, and into 256 in the two dimensional case. Using this algorithm we performed numerical calculations for the chemotactic equations. As the results, the following two things were clarified mainly on the global behavior of solutions. With the forms of the sensitive function included in the equations the pattern of cellular mold obtained from the solution changes substantially. For the chemotactic equations having the growing term the desired types of patterns of mold, that is concentric circles and ramifications, are really observed in the solutions of the equations.