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.
翻译:在本文中,我们使用一种隐含的双向延迟修正时间分解法,并与不连续的加列尔金光谱元件方法的空间分解法相结合,以解决(非)线性PDE。由此产生的数字方法在空间和时间上高度精确。由于新方案处理两种时间衍生物,必须界定两种衍生物的空间操作员。这导致该办法的扩大系统矩阵。我们分析了这个矩阵,其中涉及可能的简化和解决新出现的(非线性)方程式系统的有效方法。它展示了精心设计的前提条件和无矩阵法方法如何使新方案得到有效的实施和应用。对于线性反动和可压缩的 Euler 方程式,都显示到时间的精确度的第八顺序。最后,它说明了该方法如何用来为可压缩的Navier-Stoks方程式提供近似解决办法。