A new energy and enstrophy conserving scheme is evaluated using a suite of test cases over the global spherical domain or bounded domains. The evaluation is organized around a set of pre-defined properties: accuracy of individual opeartors, accuracy of the whole scheme, conservation, control of the divergence variable, representation of the energy and enstrophy spectra, and simulation of nonlinear dynamics. The results confirm that the scheme is between the first and second order accurate, and conserves the total energy and potential enstrophy up to the time truncation errors. The scheme is capable of producing more physically realistic energy and enstrophy spectra, indicating that the new scheme can help prevent the unphysical energy cascade towards the finest resolvable scales. With an optimal representation of the dispersive wave relations, the scheme is able to keep the flow close to being non-divergent, maintain the geostrophically balanced structures with large-scale geophysical flows over long-term simulations.
Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations
Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；BCAM - Basque Center for Applied Mathematics（巴斯克应用数学中心）；Ikerbasque, Basque Foundation for Science（伊克拉巴斯科，巴斯克科学基金会）
Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications
本文提出了一类新的求解动力学方程的守恒型半拉格朗日格式。它们是基于[S. Y. Cho等，动力学方程的保守半拉格朗日格式第一部分：重建，2020]。该方法在空间和时间上都具有高阶精度。由于半拉格朗日性质，时间步长不受CFL型条件的限制。介绍了它在刚体转动、VlasovPoisson系统和稀薄气体动力学BGK模型中的应用。在前两种情况下，采用算子分裂，以获得时间上的高阶精度，并使用保留函数的最大值和最小值的保守重建。特别是对于初始正解，这保证了L1范数的精确保持。通过将守恒重构与碰撞项的守恒处理相耦合，构造了BGK模型的守恒格式。高阶的时间是通过龙格库塔或BDF时间离散的方程沿特征线。由于L-稳定性和精确守恒，所得到的BGK模型的方案是渐近保持的基本流体动力学限制。在一维和二维空间的几个测试用例确认的方法的准确性和鲁棒性，以及AP属性的计划时，适用于BGK模型。
