In this work, we determine the full expression of the local truncation error of hyperbolic partial differential equations (PDEs) on a uniform mesh. If we are employing a stable numerical scheme and the global solution error is of the same order of accuracy as the global truncation error, we make the following observations in the asymptotic regime, where the truncation error is dominated by the powers of $\Delta x$ and $\Delta t$ rather than their coefficients. Assuming that we reach the asymptotic regime before the machine precision error takes over, (a) the order of convergence of stable numerical solutions of hyperbolic PDEs 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. We have tested our theory against numerical methods employing Method of Lines and not against ones that treat space and time together, and we have not taken into consideration the reduction in the spatial and temporal orders of accuracy resulting from slope-limiting monotonicity-preserving strategies commonly applied to finite volume methods. Otherwise, our theory applies to 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. If the PDE is reduced to an ordinary differential equation (ODE) by specifying the spatial gradients of the dependent variable and the coefficients and the source terms to be zero, then the standard local truncation error of the ODE is recovered. We perform the analysis with generic and specific hyperbolic PDEs using the symbolic algebra package SymPy, and conduct a number of numerical experiments to demonstrate our theoretical findings.
翻译:在此工作中, 我们确定在统一网格中双曲部分偏差方程式( PDEs) 的本地脱轨错误的完整表达。 如果我们使用一个稳定的数值公式, 而全球解决方案错误的精确度与全球脱轨错误的精确度相同, 我们在无线制度下做以下观察, 脱轨错误以 $\ Delta x$ 和 $\ Delta t$ 而不是其系数为主, 假设我们在机器精确度错误发生之前就到达了无线多线差校正制度, (a) 双曲PDEs 的稳定的数值解决方案的趋同顺序与 $\ Delta t$ 到 $\ Delta x$ 的精确率相同, 我们在无线差差差差差差差的最小值中进行以下观察: 使用线性方法, 我们的数值方法, 而不是用处理空间和时间的相同错误, 我们没有考虑在 平流离轨的平流和直径直径直径分析中, 将空间和直径直径的直径直径直径直径直径直到 。