In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method is high order accurate in space and time. As the novel scheme handles two time derivatives, the spatial operator for both derivatives has to be defined. This results in an extended system matrix of the scheme. We analyze this matrix regarding possible simplifications and an efficient way to solve the arising (non-)linear system of equations. It is shown how a carefully designed preconditioner and a matrix-free approach allow for an efficient implementation and application of the novel scheme. For both, linear advection and the compressible Euler equations, up to eighth order of accuracy in time is shown. Finally, it is illustrated how the method can be used to approximate solutions to the compressible Navier-Stokes equations.

中文翻译：

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非连续伽辽金法的二阶导数递延校正时间离散化

在本文中，我们使用隐式二导数延迟校正时间离散化方法，并将其与不连续伽辽金谱元方法的空间离散化相结合来求解（非线性）线性偏微分方程。由此产生的数值方法在空间和时间上都是高阶精确的。由于新方案处理两个时间导数，因此必须定义两个导数的空间算子。这导致该方案的扩展系统矩阵。我们分析这个矩阵的可能的简化和有效的方法来解决出现的（非线性）线性方程组。展示了精心设计的预处理器和无矩阵方法如何有效实现和应用新方案。对于线性平流和可压缩欧拉方程，显示了高达八级的时间精度。最后，说明了如何使用该方法来逼近可压缩 Navier-Stokes 方程的解。