双曲方程谱元有限体积法的构造分析和应用

本项目研究双曲方程新谱元有限体积法的构造和理论分析。在构造方面，将基于新加细剖分策略和广义Riemann问题建立具有最优收敛阶的新型谱元有限体积格式。在理论分析方面，除了分析新格式的稳定性和收敛性以外，还将研究新格式的L2稳定和误差估计以及某些超收敛性质。本项目的构造和分析，不只针对单个线性双曲方程，还将用于若干非线性双曲方程组如浅水波方程，电磁方程，Euler方程以及Navier-Stokes方程等。. 本项目研究构想的提出与申报者前一个自然科学基金资助项目“高次有限体积法的构造，理论分析和应用”有关。我们将把在上一项目研究中获得的面向椭圆型方程的算法构造思想和理论分析技巧应用到双曲方程。本项目将丰富和发展有限体积法理论，为与双曲方程有关的各物理学科提供数值算法支撑。

In this project, we study the design and analysis of novel spectral finite volume methods for hyperbolic equations. On the design, we will construct new spectral finite volume schemes based on novel mesh-refinement strategy and generalized Riemann problems. On the analysis, besides the normal study on the stability and accuracy, we will also estimate the L2-norm error and study the superconvergence properties of our new spectral finite volume schemes as well. The novel spectral finite volume schemes will be designed and analyzed not only for linear scalar hyperbolic equation, but also for nonlinear hyperbolic equation systems such as shallow water equations，electromagentical equations，Euler equations and Navier-Stokes equations as well...The proposed project is related to our previous NSFC program "The construction, analysis and applications of some high order finite volume methods". Some design ideas and analysis techniques, mainly for elliptic equations in the previous NSFC program, will be applied to hyperbolic equations in this project. The study of this project will develop the theory of finite volume methods and support the study of related physical disciplines.