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A unified discontinuous Galerkin framework for time integration

机译：统一的不连续Galerkin框架用于时间积分

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We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time-stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time-stepping schemes, such as low-order one-step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time-stepping schemes.
机译：我们引入了一种新的不连续Galerkin方法进行时间积分。在加权残差法的基础上，在有限元时间离散化中采用数值正交来解决一般的非线性常微分方程。在变分分析中，对测试函数强制执行许多不同的条件，包括显式，隐式和辛条件，以获得所需的时间步进方案特征。所提出的不连续Galerkin方法提供了一个统一的框架来导出各种时间步长方案，例如低阶一步法，Runge-Kutta方法和多步法。在提出的框架的基础上，构造了几种不同阶的显式Runge-Kutta方法。辛Runge-Kutta方法的推导也已经实现。所提出的框架允许在一些特性（例如准确性，稀疏性和稳定性）方面优化新方案。精度优化是基于线性测试初始值问题的误差估计函数的解析形式进行的。发现具有较高形式精度的方案可以提供更准确的解决方案。我们还探索了稀疏性的优化潜力，这与信号/成像处理中的一般压缩感测有关。稳定性区域的两个关键维度，即沿假想轴和负实轴的最大间隔，被用作稳定性优化的标准。这给出了分别求解双曲和抛物型偏微分方程的最大库兰特-弗里德里希斯-路易​​时间步长。进行数值实验以验证优化的时间步长方案。

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