A New Class of Runge-Kutta Methods for Nonlinearly Partitioned Systems

Abstract: 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.