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.

翻译：对一种显式半拉格朗日谱元方法的相位误差和耗散误差进行了离散分析。该半拉格朗日方法根据输运方程的拉格朗日形式平流拉格朗日插值，并采用最小二乘拟合来修正相邻单元界面约束的更新。通过假设采用单项式表示而非拉格朗日形式，推导了单个单元上算法的离散版本。由此产生的代数系统适用于修正方程分析和特征值分析。修正方程分析在单一空间位置和时间实例上对模板进行泰勒展开，表明在单元尺寸趋近于零的极限下，半拉格朗日方法与输运方程的偏微分方程形式一致。修正方程的首阶截断项与插值多项式的阶数同阶，这与文献中报告的数值测试结果一致。色散关系表明，该方法色散可忽略不计，这是半拉格朗日方法的常见特征。特征值分析表明，采用节点切比雪夫插值的半拉格朗日方法在基于单元内最小配置节点间距的Courant-Friedrichs-Lewy条件大于1时是稳定的。