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

TL;DR

This work introduces nonlinearly partitioned Runge-Kutta (NPRK) methods for solving IVPs of the form $y' = F(y,y)$, enabling different implicitness levels to be applied to each nonlinear argument. NPRK generalizes classical RK, additive ARK, and partitioned PRK methods, and connects to SIRK while supporting more flexible implicitness patterns, including IMEX and IMIM variants. The paper analyzes linear stability, derives order conditions via underlying PRK structures, and provides concrete IMEX and IMIM NPRK schemes with up to five stages and orders 2–3, complemented by a practical sequential algorithm. Numerical experiments on Burgers' equation and gray thermal radiation transport illustrate that NPRK delivers high-order accuracy, large-stencil stability, and reduced implicit solves compared to fully implicit schemes, offering a scalable approach for nonlinearly coupled, stiff systems. The framework promises broad applicability to multiscale PDE/PIDE models where standard additive or component partitioning is insufficient.

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.

Figures (6)

• Figure 1: Convergence diagrams for the viscous Burgers' equation \ref{['eq:burgers']}. We discretized the domain $x \in [-2,2]$ using 1000 spatial grid points, and integrated to time $t=6/10$ using the initial condition $u(x,t=0) = e^{-3x^2}$.
• Figure 1: Linear stability region slices for IMEX-NPRK 2[31] (top row), 2[32]a or 2[42]a (middle row), and 2[32]b or 2[42]b (bottom row). Each contour is a two-dimensional slice $\mathcal{P}(z_1)$, from \ref{['eq:stability-region-symmetric']}, with $\arg(z_1) \in \{\pi, \frac{3\pi}{4}, \frac{\pi}{2}\}$ and $|z_1| \in \{0,1,\ldots,10\}$; contours corresponding to $|z_1|=0,5,10$ are colored black, red, and blue, respectively. These choices of $\arg(z_1)$ respectively approximate an implicit linear component with diffusion, a mix of diffusion and oscillation, and pure oscillation. The regions shaded in grey are $\widetilde{\mathcal{P}}(\arg(z_1))$ from \ref{['eq:stability-region-symmetric-max-r-theta']}.
• Figure 1: Convergence diagrams for the viscous Burgers' equation \ref{['eq:burgers']} using the nonlinear partitions \ref{['eq:burgers-nonlinear-partitioning-non-conservative', 'eq:burgers-nonlinear-partitioning-conservative']} . We discretized the domain $x \in [-8,8]$ using 1000 spatial grid points, and integrate to time $t=20$ using the initial condition $u(x,t=0) = e^{-3x^2}$.
• Figure 2: Propagation of Marshak wave from initially uniform background temperature with coarse spatial mesh at four subsequent times in sh (taken from imex-trt).
• Figure 3: Propagation of Marshak wave from initially uniform background temperature solved using a fine spatial grid (top) and coarse spatial grid (bottom). All plots show relative error in radiation energy.

Theorems & Definitions (16)

• Definition 1: Underlying method
• Definition 2
• Proposition 3
• Proof 1
• Proposition 4
• Proof 2
• Remark 1
• Remark 2
• Remark 3
• Proposition 1