The Rosenbrock–Krylov family of time integration schemes is an extension of Rosenbrock-W methods that employs a specific Krylov based approximation of the linear system solutions arising within each stage of the integrator. This work proposes an extension of Rosenbrock–Krylov methods to address stability questions which arise for methods making use of inexact linear system solution strategies. Two approaches for improving the stability and efficiency of Rosenbrock–Krylov methods are proposed, one through direct control of linear system residuals and the second through a novel extension of the underlying Krylov space to include stage right hand side vectors. Rosenbrock–Krylov methods employing the new approaches show a substantial improvement in computational efficiency relative to prior implementations.

Rosenbrock-Krylov系列时间积分方案是Rosenbrock-W方法的扩展，它采用了特定的基于Krylov的积分器各个阶段内线性系统解决方案的近似值。这项工作提出了Rosenbrock-Krylov方法的扩展，以解决因使用不精确的线性系统求解策略而引起的稳定性问题。提出了两种提高Rosenbrock-Krylov方法的稳定性和效率的方法，一种是通过直接控制线性系统残差，另一种是通过对基础Krylov空间进行新颖的扩展以包括阶段右侧向量。采用新方法的Rosenbrock-Krylov方法相对于先前的实现方式显示出计算效率的显着提高。