Order Conditions for Nonlinearly Partitioned Runge-Kutta Methods

• DOI: 10.48550/arxiv.2401.12427
• Publication Date: 2024-01-01
• Authors: Brian K. Tran; Ben S. Southworth; Tommaso Buvoli
• Keywords: Ode; Tree (set theory); Runge–Kutta methods; Colored

Recently a new class of nonlinearly partitioned Runge-Kutta (NPRK) methods was proposed for systems ordinary differential equations, $y' = F(y,y)$. The target problems are ones in which different scales, stiffnesses, or physics coupled nonlinear way, wherein the desired partition cannot be written classical additive component-wise fashion. Here we use rooted-tree analysis to derive full order conditions NPRK$_M$ methods, where $M$ denotes number partitions. Due coupling and thereby mixed product differentials, it turns out standard node-colored used analyzing ODE integrators does not naturally apply. Instead develop edge-colored framework address coupling. resulting enumerated, provided directly up 4th with $M=2$ 3rd-order $M=3$, related existing RK methods.