Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of equations to solve for each time step. These algebraic systems couple all Runge-Kutta stages together, giving a much larger system than for single-stage methods. We consider an approach to these systems based on monolithic smoothing. If stage-coupled smoothers possess a certain kind of structure, then the question of convergence of a two-grid or multi-grid iteration reduces to convergence of a related strategy for a single-stage system with a complex-valued time step. In addition to providing a general theoretical approach to the convergence of monolithic multigrid methods, several numerical examples are given to illustrate the theory show how higher-order Runge-Kutta methods can be made effective in practice.

翻译：时间相关问题的有限元离散化需要有效的时间步进方案。虽然隐式Runge-Kutta方法具有优越的精度和稳定性，但每个时间步需要求解庞大且复杂的方程组。这些代数系统将所有Runge-Kutta阶段耦合在一起，生成比单阶段方法规模大得多的系统。本文考虑基于单体平滑的求解策略。若阶段耦合平滑器具备特定结构，则两网格或多网格迭代的收敛性问题可转化为对带有复时间步长的单阶段系统相关策略的收敛性分析。除提供单体多重网格方法收敛性的通用理论框架外，文中还给出多个数值算例以阐明理论，展示高阶Runge-Kutta方法在实际应用中的有效性。