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

小行星9805748 拟线性反应扩散对流微分方程组 方程在模拟物理现象中具有广泛的重要性， 如生物学、化学、冶金学和燃烧学等不同领域。 一些例子是流体中的剪切流， 粘度;神经束的Hodgkin-Huxley型模型;以及 疾病在人群中的传播。 因为这些问题是 典型地高度非线性并且表现出复杂的行为、数值 微分方程的解已经成为 调查他们的财产。 然而，这些相同的品质 提出了一些基本的科学问题：如何才能可靠， 准确估计信息的准确性， 如何获得数值解，以及如何获得所需的精度范围 实现？ 该项目将从三个层次来解决这些问题：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发展一个后验理论的错误 用数量来估计误差的估计 取决于可以计算的数值解， 近似。然后，该理论将用于开发以下方法： 计算误差估计和自适应误差控制。PI还将分析计算误差估计的可靠性和准确性，并且由于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属性的保持。这个项目的实际目标包括：一个公共访问的代码，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， 以上所述，以及教育部分的形式， 基本非线性模型的有限元法教科书 科学

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.