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Hyperbolic problems with discontinuous coefficients

具有不连续系数的双曲问题

基本信息

批准号：
EP/V005529/1
负责人：
Claudia Garetto
金额：
$ 74.8万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV005529%2F1
关键词：
Hyperbolic problems discontinuous coefficients

项目摘要

Linear and nonlinear hyperbolic PDEs arise in all sciences (physics, chemistry, medicine, engineering, astronomy, etc). In particular, in physics they model several important phenomena, from propagation of waves in a medium (for instance propagation of seismic waves during an earthquake) to refraction in crystals and gas-dynamics. When modelling wave propagation trough a multi-layered medium, for instance the subsoil during an earthquake, it is physically meaningful to make use of discontinuous functions. This project wants to study the largest possible class of hyperbolic equations and systems: with variable multiplicities and discontinuous coefficients (depending on time and space). This is notoriously a very difficult problem due to the presence of multiplicities and the low-regularity of the coefficients. It will require the development of new analytical methods which will be first introduced under assumptions of regularity (first part of the project) and then gradually adapted to less regular coefficients (second part of the project). In order to provide a unified approach to hyperbolic problems with discontinuous coefficients, we will test the strength of our new analytical methods numerically. This will build a bridge between two different approaches to hyperbolic PDEs (analytical and numerical), a bridge based on analysis, comparison and implementation of new ideas.

线性和非线性双曲偏微分方程出现在所有科学（物理、化学、医学、工程、天文学等）中。特别是，在物理学中，它们模拟了几种重要的现象，从波在介质中的传播（例如地震中地震波的传播）到晶体和气体动力学中的折射。当模拟波在多层介质中的传播时，例如地震时的底土，使用不连续函数在物理上是有意义的。这个项目想要研究最大可能的双曲方程和系统：具有可变多重性和不连续系数（取决于时间和空间）。由于存在多重性和系数的低正则性，这是一个非常困难的问题。这将需要发展新的分析方法，这些方法将首先在规则假设下引入（项目的第一部分），然后逐渐适应不那么规则的系数（项目的第二部分）。为了给具有不连续系数的双曲型问题提供一个统一的方法，我们将用数值方法来检验我们的新分析方法的强度。这将在双曲偏微分方程的两种不同方法（解析和数值）之间建立一座桥梁，一座基于分析、比较和实施新思想的桥梁。