AI 中文总结
该研究针对Vlasov-Poisson系统，采用非对称加权Hermite基的谱方法，通过Galerkin方法解决其不稳定性问题，提出等价形式降低计算成本，数值模拟验证了方法的稳定性。
AI 中文摘要
我们研究一种针对Vlasov-Poisson（VP）系统的数值方法，该方法利用速度空间中的非对称加权（AW）Hermite基，VP系统是一个双曲系统。特别地，我们聚焦于速度空间中的谱方法。对于VP系统的Hermite谱形式，我们分析了采用AW Hermite基的形式可能不稳定的原因。为获得L2稳定性，我们采用Galerkin方法而非经典的Petrov-Galerkin方法，该方法自然保证了关于L2范数的稳定性。我们还提出了该方法的等价形式，其计算成本与Petrov-Galerkin方法相比保持适度。最后，我们基于所提出的Hermite谱方法开展数值模拟，展示了其稳定性。
英文摘要
We investigate a numerical method for the Vlasov-Poisson (VP) system utilizing asymmetrically-weighted (AW) Hermite bases in velocity space, which is an hyperbolic system. In particular, we concentrate on spectral methods in velocity. For the Hermite spectral form of the VP system, we analyze the resaon that the form with AW Hermite bases can be instable. To obtain L2 stability properties, we consider a Galerkin method intead of the classical Petrov-Galerkin method, which naturally ensures stability with respect to the L2 norm. We also present an equivalent form of the method that maintains a computational cost modest compared to that of the Petrov-Galerkin method. Finally, we present numerical simulations based on the proposed Hermite spectral method, showcasing its stability.
