This work introduces a new class of Runge-Kutta methods for solving nonlinearly partitioned initial value problems. These new methods, named nonlinearly partitioned Runge-Kutta (NPRK), generalize existing additive and component-partitioned Runge-Kutta methods, and allow one to distribute different types of implicitness within nonlinear terms. The paper introduces the NPRK framework and discusses order conditions, linear stability, and the derivation of implicit-explicit and implicit-implicit NPRK integrators. The paper concludes with numerical experiments that demonstrate the utility of NPRK methods for solving viscous Burger's and the gray thermal radiation transport equations.

翻译：本文提出了一类新的龙格-库塔方法，用于求解非线性分区初值问题。这类新方法被命名为非线性分区龙格-库塔（NPRK）方法，它推广了现有的加性分区龙格-库塔方法和分量分区龙格-库塔方法，允许在非线性项内分配不同类型的隐式性。本文介绍了NPRK框架，讨论了阶条件、线性稳定性，并推导了隐式-显式及隐式-隐式NPRK积分器。最后，通过数值实验展示了NPRK方法在求解粘性伯格斯方程和灰体热辐射输运方程中的实用性。