论文标题
多级光谱递延校正的收敛分析
Convergence analysis of multi-level spectral deferred corrections
论文作者
论文摘要
光谱递延校正（SDC）方法是用于普通微分方程（ODE）的迭代求解器类。它可以解释为搭配问题的预处理PICARD迭代。该方法的收敛性是众所周知的，对于合适的问题，它在迭代中获得了一个顺序，直到提供了搭配问题的正交方法的顺序。这项吸引人的功能可轻松创建用于ODE的灵活，高阶精确方法。 SDC的变化是多级光谱递延校正（MLSDC）。在这里，迭代是在级别和FAS校正项的层次结构上进行的，如非线性多机方法，夫妻在不同级别上解决方案。尽管有几个数字示例显示其功能和效率，但仍缺少理论融合证明。本文解决了这个问题。将给出MLSDC收敛性的证明，包括确定时间步长的收敛速率，并将在数值上证明理论分析的结果。事实证明，该方法比SDC对收敛率有限制。
The spectral deferred correction (SDC) method is class of iterative solvers for ordinary differential equations (ODEs). It can be interpreted as a preconditioned Picard iteration for the collocation problem. The convergence of this method is well-known, for suitable problems it gains one order per iteration up to the order of the quadrature method of the collocation problem provided. This appealing feature enables an easy creation of flexible, high-order accurate methods for ODEs. A variation of SDC are multi-level spectral deferred corrections (MLSDC). Here, iterations are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. While there are several numerical examples which show its capabilities and efficiency, a theoretical convergence proof is still missing. This paper addresses this issue. A proof of the convergence of MLSDC, including the determination of the convergence rate in the time-step size, will be given and the results of the theoretical analysis will be numerically demonstrated. It turns out that there are restrictions for the advantages of this method over SDC regarding the convergence rate.
