Semi-implicit-explicit Runge-Kutta method for nonlinear differential equations
Lingyun Ding
TL;DR
The paper addresses the computational burden of IMEX schemes for stiff PDE-inspired ODEs by introducing semi-IMEX Runge-Kutta methods that treat the stiff term $G(t,\mathbf{u})\mathbf{u}$ explicitly while solving only linear systems in the implicit part. It develops a hierarchy of schemes with first-, second-, and third-order accuracy, analyzes stability properties (including $A$- and $L$-stability) and row-sum conditions, and provides explicit Butcher tableaux. Numerical tests on scalar equations, nonlinear diffusion, Cahn–Hilliard, and Navier–Stokes demonstrate accurate convergence and significantly larger stable time steps compared to IMEX with linear splitting, underscoring the practical efficiency and applicability of the approach. A public repository accompanies the work to facilitate reproduction and further development, with future directions including higher-order methods and alternative stability notions.
Abstract
A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where $\mathbf{f}$ is a non-stiff term and $G\mathbf{u}$ represents the stiff terms. Such systems frequently arise from spatial discretizations of time-dependent nonlinear partial differential equations (PDEs). For instance, $G$ could involve higher-order derivative terms with nonlinear coefficients. Traditional IMEX-RK methods, which treat $\mathbf{f}$ explicitly and $G\mathbf{u}$ implicitly, require solving nonlinear systems at each time step when $G$ depends on $\mathbf{u}$, leading to increased computational cost and complexity. In contrast, the proposed semi-IMEX scheme treats $G$ explicitly while keeping $\mathbf{u}$ implicit, reducing the problem to solving only linear systems. This approach eliminates the need to compute Jacobians while preserving the stability advantages of implicit methods. A family of semi-IMEX RK schemes with varying orders of accuracy is introduced. Numerical simulations for various nonlinear equations, including nonlinear diffusion models, the Navier-Stokes equations, and the Cahn-Hilliard equation, confirm the expected convergence rates and demonstrate that the proposed method allows for larger time step sizes without triggering stability issues.