In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the coefficients of the generic global truncation error. Our analysis is valid for any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method, belonging to the Method of Lines. Furthermore, we discuss the practical implications of this analysis. If we employ a stable numerical scheme and the orders of accuracy of the global solution error and the global truncation error agree, we make the following asymptotic observations: (a) the order of convergence at constant ratio of $\Delta t$ to $\Delta x$ is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. An implication of (a) is that it is impractical to invest in a time-stepping method of order higher than the spatial discretization. In addition to (b), we demonstrate that under certain circumstances, the error can even monotonically increase with refinement only in space or only in time, and explain why this phenomenon occurs. To verify our theoretical findings, we conduct convergence studies of linear and non-linear advection equations using finite difference and finite volume spatial discretizations, and predictor-corrector and multistep time-stepping methods. Finally, we study the effect of slope limiters and monotonicity-preserving strategies on the order of accuracy.
翻译:在这项工作中,我们确定双曲部分偏差方程式(PDEs)的全球脱轨错误的完整表达式。特别是,我们使用理论分析和象征性代数来寻找通用全球脱轨差数系数的精确表达式。我们的分析适用于任何双曲PDE,无论是线性还是非线性,并使用空间中有限差数、有限体积或有限元素分解,以及属于“线条方法”的预测器-校正、多步制或推迟校正方法。此外,我们讨论这一分析的实际影响。如果我们使用稳定的数值和全球解析差差差差差差差系数的精确排序。我们的分析适用于任何双曲PDE,无论是线性还是非线性,并且使用一定的差差差差差差差值、多步制或延迟校正法方法。(a)在时间或时间精确度研究中,我们只能保证不精确的趋同,在时间精确度研究中,我们只能用某种不切实际的方法来解释。(a)在时间精确的精确度上,在精确度上,我们只能用某种方法来解释。