Numerical Amalysis and Glolal Behavioz of Chemotactic Equations

1998年我们设计了一种基于有限元法和Runge-Kutta方法的趋化方程离散格式，并从理论上证明了该格式的稳定性和近似解的收敛性。在证明过程中，引入了离散半群和离散演化算子的概念，精确地描述了近似解，并在1999年建立了算法。对于某些有限元方法，如果采用常规算法，则需要大量的计算机内存。因此，在这项研究中，我们做了一些设备，我们交换矩阵的成分以适当的方式，以压缩矩阵的频带的大小。由此，空间变量在一维情况下可以被划分为8192，在二维情况下可以被划分为256。使用该算法，我们进行了数值计算的趋化方程。作为结果，以下两件事主要澄清的整体行为的解决方案。随着方程中包含的敏感函数的形式，从解中得到的细胞模型的图案发生了很大的变化。对于具有增长项的趋化方程，在方程的解中确实观察到了所需类型的模具图案，即同心圆和分枝。

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.