论文标题
分析明确的高阶半拉格朗日节点方法
Analysis of an Explicit, High-Order Semi-Lagrangian Nodal Method
论文作者
论文摘要
对明确的半拉格朗日光谱元素方法的相位和耗散误差进行离散分析。半拉格朗日方法按照传输方程的拉格朗日形式对拉格朗日的插值进行递增，并使用最小二乘拟合来纠正相邻元素接口约束的更新。通过假设单一表示而不是Lagrange形式，可以得出单个元素上算法的离散版本。由此产生的代数系统既可以进行修改的方程分析和特征值分析。 Taylor在单个空间位置和时间实例上扩展模板的修改方程分析表明，半拉格朗日方法与传输方程的PDE形式在元素大小为零的极限中的PDE形式一致。修改方程的领先顺序截断项是插值度的程度的顺序，这与文献中报道的数值测试一致。分散关系表明该方法是分散的，对于半拉格朗日方法来说是常见的。特征值分析表明，具有淋巴结chebyshev interpolant的半拉格朗日方法对于基于最小搭配节点间距的courant-friedrichs-lewy条件是稳定的，该条件大于统一。
A discrete analysis of the phase and dissipation errors of an explicit, semi-Lagrangian spectral element method is performed. The semi-Lagrangian method advects the Lagrange interpolant according the Lagrangian form of the transport equations and uses a least-square fit to correct the update for interface constraints of neighbouring elements. By assuming a monomial representation instead of the Lagrange form, a discrete version of the algorithm on a single element is derived. The resulting algebraic system lends itself to both a Modified Equation analysis and an eigenvalue analysis. The Modified Equation analysis, which Taylor expands the stencil at a single space location and time instance, shows that the semi-Lagrangian method is consistent with the PDE form of the transport equation in the limit that the element size goes to zero. The leading order truncation term of the Modified Equation is of the order of the degree of the interpolant which is consistent with numerical tests reported in the literature. The dispersion relations show that the method is negligibly dispersive, as is common for semi-Lagrangian methods. An eigenvalue analysis shows that the semi-Lagrangian method with a nodal Chebyshev interpolant is stable for a Courant-Friedrichs-Lewy condition based on the minimum collocation node spacing within an element that is greater than unity.
