A new preconditioner based on a block $LDU$ factorization with algebraic multigrid subsolves for scalability is introduced for the large, structured systems appearing in implicit Runge-Kutta time integration of parabolic partial differential equations. This preconditioner is compared in condition number and eigenvalue distribution, and in numerical experiments with others in the literature: block Jacobi, block Gauss-Seidel, and the optimized block Gauss-Seidel method of Staff, Mardal, and Nilssen [{\em Modeling, Identification and Control}, 27 (2006), pp. 109-123]. Experiments are run on two test problems, a $2D$ heat equation and a model advection-diffusion problem, using implicit Runge-Kutta methods with two to seven stages. We find that the new preconditioner outperforms the others, with the improvement becoming more pronounced as spatial discretization is refined and as temporal order is increased.

翻译：对于在抛物线部分差分方程中隐含的龙格-库塔时间整合中出现的大型结构化系统,采用了基于代数多格多格子溶解法的以方块 $LDU$乘数计算法的新先决条件。该先决条件在条件数和电子值分布方面以及与文献中其他人的数值实验中进行了比较:Jacobi区块、Gaus-Seidel区块、以及工作人员最优化的Gaus-Seidel区块法、Mardal和Nilsen[#em 建模、识别和控制],27(2006年),第109-123页。实验是在两个测试问题上进行的,一个是2D$热方程,另一个是模型的反位扩散问题,使用2至7个阶段的内置Runge-Kutta方法。我们发现,新的先决条件优于其他方法,随着空间离散化的更加明显,随着时间顺序的提高,我们发现新的改进情况比其他方法要好。