We randomize the implicit two-stage Runge-Kutta scheme in order to improve the rate of convergence (with respect to a deterministic scheme) and stability of the approximate solution (with respect to the solution generated by the explicit scheme). For stability analysis, we use Dahlquist's concept of A-stability, adopted to randomized schemes by considering three notions of stability: asymptotic, mean-square, and in probability. The randomized implicit RK2 scheme proves to be A-stable asymptotically and in probability but not in the mean-square sense.

翻译：为提升近似解的收敛速度（相对于确定性格式）及稳定性（相对于显式格式生成的解），我们对隐式两级龙格-库塔格式进行随机化处理。在稳定性分析中，采用达尔奎斯特（Dahlquist）所提出的A-稳定性概念，并通过渐近稳定性、均方稳定性及依概率稳定性三种定义将其适配至随机格式。经证明，随机隐式RK2格式在渐近意义与依概率意义下具有A-稳定性，但在均方意义下不成立。