Order Conditions for Nonlinearly Partitioned Runge-Kutta Methods

TL;DR

This paper develops a comprehensive framework for order conditions of nonlinearly partitioned Runge--Kutta (NPRK) methods solving y' = F(y,y). By introducing edge-colored rooted trees, it derives full NPRK_M order conditions, showing they differ from traditional ARK conditions due to nonlinear coupling; it provides explicit condition sets up to 4th order for M=2 and 3rd order for M=3, along with algorithms to enumerate them. The work clarifies the relationship between NPRK and additive ARK/PRK methods, highlighting new nonlinear coupling conditions that vanish under additively partitioned problems. A numerical example on Lotka--Volterra demonstrates how the nonlinear order conditions yield embedded estimates of state-dependent coupling strength, illustrating practical use in multiphysics systems. The results pave the way for constructing high-order NPRK methods with controlled nonlinear coupling and offer tools for assessing and exploiting nonlinear interactions in partitioned dynamics.

Abstract

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

Figures (3)

• Figure 1: Convergence test for Method 1 and Method 2 for $\alpha = 0, 0.01, 2$ with $u(0)=1=v(0)$ and final time $T=1$. The reference solution was computed using Method 1 with $h = 10^{-4}$. The slopes were determined using linear regression.
• Figure 2: Embedded estimate \ref{['eq:nonlinear-coupling-strength']} of nonlinear coupling after one step as a function of $\alpha$ for several choices of step size $h = 10^{-2}, 10^{-3}, 10^{-4}$.
• Figure 3: Asymptotic scaling of the embedded estimate \ref{['eq:nonlinear-coupling-strength']} with respect to $h$ for various choices of $\alpha$. The slopes were determined using linear regression.

Theorems & Definitions (15)

• Example 1.1
• Definition 2.1
• Definition 2.2
• Proposition 3.1
• Theorem 3.1: NPRK Order Conditions
• proof
• Remark 3.1
• Theorem 3.2: NPRK$_M$ Order Conditions
• Remark 3.2
• Remark 3.3