论文标题
runge的错误分析 - 线性时间相关的偏微分方程的kutta不连续的盖尔金方法
Error analysis of Runge--Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations
论文作者
论文摘要
在本文中，我们提出了完全离散的Runge-Kutta不连续的Galerkin（DG）方案的误差估计值，以实现线性时间相关的偏微分方程。该分析适用于任何订单的显式runge-kutta时间离散。对于空间离散化，考虑了一般离散操作员，其中涵盖了各种DG方法，例如上风偏置的DG方法，中央DG方法，局部DG方法和超湿DG方法。如果解决方案足够光滑，并且存在具有某些属性的空间操作员，我们将获得稳定且一致的完全离散方案的误差估计。讨论了对双曲线保护定律，热方程，分散方程和波动方程的应用。特别是，我们提供了局部DG方法的最佳误差估计值的替代证明，以在一个维度中具有高阶衍生物的方程式，该方程不依赖辅助未知数的能量不等式。
In this paper, we present error estimates of fully discrete Runge--Kutta discontinuous Galerkin (DG) schemes for linear time-dependent partial differential equations. The analysis applies to explicit Runge--Kutta time discretizations of any order. For spatial discretization, a general discrete operator is considered, which covers various DG methods, such as the upwind-biased DG method, the central DG method, the local DG method and the ultra-weak DG method. We obtain error estimates for stable and consistent fully discrete schemes, if the solution is sufficiently smooth and a spatial operator with certain properties exists. Applications to schemes for hyperbolic conservation laws, the heat equation, the dispersive equation and the wave equation are discussed. In particular, we provide an alternative proof of optimal error estimates of local DG methods for equations with high order derivatives in one dimension, which does not rely on energy inequalities of auxiliary unknowns.
