Mathematical Sciences: Analytical and Numerical Studies of Nonlinear Hyperbolic Systems

本项目从分析和数值两个角度研究非线性双曲型偏微分方程组，重点研究等离子体物理和Vlasov-Maxwell系统的问题。我们将研究高维系统的质点方法，以获得收敛和高阶精度为目标。碰撞效应(例如来自朗道系统的碰撞效应)也将被包括在内。目前使用的许多数值格式几乎没有(如果有的话)理论上的合理性。对于那些有充分证据的少数人来说，已知的准确性相对较差。因此，从实际角度来看，有很多改进是可能的。此外，目前还没有关于碰撞效应的严格研究(无论是理论上的还是计算上的)。这些研究将在聚变能源等领域得到应用。

This project involves the study of nonlinear hyperbolic systems of partial differential equations from both analytical and numerical points of view, with emphasis on problems from Plasma Physics and the Vlasov-Maxwell system. Particle Methods for higher-dimensional systems will be studied, with the goal of obtaining convergence as well as higher-order accuracy. Collisional effects (e.g. those from the LANDAU system) will also be included. Many of the presently-used numerical schemes have little (if any) theoretical justification. For those few that are well founded, the known accuracy is relatively poor. Thus from a practical point of view, much improvement is possible. Moreover, at this time there are no rigorous studies (neither theoretical nor computational) of collisional effects. These studies will have application in areas such as fusion energy.