几类延迟偏微分方程数值解法研究

As we know, fixed mesh methods are not well suited to the numerical simulations of blow-up problems. So, in this study, we develop high-accuracy adaptive grid methods for blow-up solution of qusilinear parabolic problems with constant delay, and establish the theories of blow-up of numerical solution. This study is also devoted to alternating direction implicit (ADI) methods for two types of parabolic equations with constant delay, which preserve the delay dependent asymptotic stability of the linear test problem under considerations. The applications of Richardson extrapolation (RE) methods to these ADI methods are carried out. And the influences of RE Methods on stability of numerical solutions are also discussed. Theoretical results of several highly accurate compact ADI methods for three kinds of parabolic equations with constant delay are first constructed. Then RE methods, which are used along with these compact ADI methods, can improve temporal accuracy. For convective-diffusion equations with two constant delays, we further develop characteristic methods, compact (block) boundary methods, and establish corresponding preconditioned iterative methods with low computational cost. Using and improving the treatments for various classes of delays used in [22-24], numerical methods stated above can be adapted to solve corresponding problems including pantograph delay, or variable delay or many variable delays, etc. Finally, improving the techniques used to treat discountious problems in [13], numerical methods stated above can be creatively extended to the simulations of discountious solutions.

针对固定网格方法不适用于爆破解的数值模拟，本课题设计拟线性常延迟抛物方程爆破解的高精度自适应网格方法，并研究数值解的爆破理论。本课题研究两类常延迟抛物方程保持渐近稳定的交替方向隐式 (ADI) 法及其 Richardson 外推法，并讨论 Richardson 外推法对数值解稳定性的影响；研究三类拟线性延迟抛物方程的高精度紧 ADI 法及其理论，并设计相应的 Richardson 外推法，提高时间精度；建立两常延迟对流扩散方程的特征线方法、紧 (块) 边值方法以及算法理论，并发展相应的预估迭代方法，以节省计算成本；利用和改进文 [22-24] 中处理复杂延迟的技术，调整上述算法使它们能适用于比例延迟、变延迟、多变延迟等情形；最后，创造性地改进文 [13] 中处理间断问题的技巧，修正上述算法使它们能适用于间断解的模拟。

在本项目里，我们建立了几类延迟偏微分方程的高效数值算法及理论. 具体成果如下： (1) 建立了模拟一类非线性延迟抛物方程爆破解的自适应网格方法及算法理论，分析了数值爆破解的爆破率，爆破区间和爆破时间等理论。(2) 建立了三类延迟抛物方程交替方向隐式法及其外推法，给出了数值解的稳定性，收敛性等结果。特别地，通过引入一些特殊时间和空间的网格剖分，构造了线性延迟抛物方程保稳定交替方向隐式法及其理论，指出了这类方法不仅有交替方向隐式法的高性能，而且具有长时间的跟踪和模拟能力。(3) 完成了一类多滞量对流扩散方程特征有限差分法、盒子格式和紧块边值法的建立和数值解的理论分析。. 此外，本项目还取得了其它结果：(1) 建立了非线性粘性波动方程两类 L^{2} 稳定的交替方向有限差分法及理论。与传统方法相比，它们有更好的稳定性，和有效抑制非物理震荡。(2) 先对一类单侧障碍问题建立第一类椭圆变分不等式， 然后运用基本解法求解了相应的变分不等式问题。 (3) 在某些假设下，运用相对重排理论，详细地证明了一类退化抛物方程重整规划解的存在性。