Computational Error Estimation and Adaptive Error Control for Numerical Solutions of Differential Equations

准线性反应-扩散-对流微分方程系统在生物学、化学、冶金和燃烧等不同领域的物理现象建模中具有广泛的重要性。一些例子是具有温度依赖粘度的流体中的剪切流动；霍奇金-赫胥黎型神经束模型；以及疾病在人群中传播的模型。由于这类问题通常是高度非线性的，表现出复杂的行为，微分方程的数值解已成为研究其性质的主要工具。然而，这些同样的特性也引起了一些基本的科学问题：如何才能可靠而准确地估计从数值解中计算出的信息的精度？如何才能达到期望的精度范围？该项目将从三个层面解决这些问题：理论分析；代码实现；并应用于实际问题。所提出的方法是基于开发一种后验误差估计理论，其中误差是根据可计算或近似的数值解的数量来估计的。该理论将用于开发计算误差估计和自适应误差控制方法。PI还将分析计算误差估计的可靠性和准确性，因为稳定性对数值解的准确性有直接影响，因此在离散化下微分方程的稳定性保持。本项目的实际目标包括：一个可公开访问的代码，用于使用计算误差估计和自适应误差控制来解决一般的反应扩散问题，应用于上面提到的实际模型，以及以教科书形式的教育部分，用于基本非线性科学模型的有限元方法。

9805748 Estep Systems of quasi-linear reaction-diffusion-convection differential equations have widespread importance in modeling physical phenomena in such diverse fields as biology, chemistry, metallurgy, and combustion. Some examples are shear flow in fluids with temperature dependent viscosity; Hodgkin-Huxley type models of nerve bundles; and models of the spread of disease in populations. Because such problems are typically highly nonlinear and exhibit complex behavior, numerical solutions of the differential equations have become the main tool for investigation of their properties. However, these same qualities give rise to some fundamental scientific questions: How can reliable and accurate estimates of the accuracy of information computed from numerical solutions be obtained and how can a desired range of accuracy be achieved? The project will address these question on three levels: theoretical analysis; implementation in code; and application to practical problems. The proposed approach is based on developing an a posteriori theory of error estimation in which the error is estimated in terms of quantities depending on the numerical solution that can be computed or approximated. The theory will then be used to develop methods of computational error estimation and adaptive error control. The PI will also analyze the reliability and accuracy of the computational error estimates and, because stability has a direct impact on the accuracy of numerical solutions, the preservation of stability properties of the differential equation under discretization. Practical goals of this project include; a publicly-accessible code for solving general reaction-diffusion problems using computational error estimation and adaptive error control, applications to the practical models like those mentioned above, and an educational component in the form of a textbook on the finite element method for the basic nonlinear models of science.