Adaptive Time-Step Semi-Implicit One-Step Taylor Scheme for Stiff Ordinary Differential Equations

TL;DR

The paper tackles stiff/non-stiff ODE integration by developing high-order Taylor-based semi-implicit and implicit one-step schemes. It derives first- and second-order SI-T-1, SI-T-2 and I-T-1, I-T-2 for the split system $U' = f(U) + g(U)$, employing explicit Jacobians where appropriate and implicit handling of the stiff part. A linear stability analysis shows $L$-stability and introduces stability regions to characterize performance, while an embedded time-step controller enables adaptive stepping. Numerical experiments on the Van der Pol problem demonstrate robustness, asymptotic preserving behavior, and competitive efficiency against ode15s and IMEX-RK(2,1), validating the practical value of the proposed methods for stiff ODEs and potentially PDEs via AP properties.

Abstract

In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a unified framework, ensuring stability and accuracy. The schemes are derived and analyzed for their consistency and stability properties, showcasing their effectiveness in practical computational scenarios.

Figures (5)

• Figure 1: Stability regions $S_1$, in the $(Re(w); Im(w))$ plane, of some second order schemes such as: IMEX-RK2 method; Implicit and Semi-Implicit Taylor method and Heun scheme.
• Figure 2: Semi-implicit numerical $y-$solution for the Van der Pol system \ref{['VDP']} with well-prepared initial conditions \ref{['IC_wp']} obtained at time $t=3\mu$ with $\Delta t_0 = 0.1$. The zoom of the second challenging boundary layers $t \approx 1.6\mu$ are reported. The reference solutions have been computed with the ode15s matlab solver.
• Figure 3: Implicit numerical $y-$solutions for the Van der Pol system \ref{['VDP']} with well-prepared initial conditions \ref{['IC_wp']} obtained at time $t=3\mu$ with $\Delta t_0 = 0.1$. The zoom of the second challenging boundary layers $t \approx 1.6\mu$ are reported. The reference solutions have been computed with the ode15s matlab solver.
• Figure 4: Semi-implicit numerical $y-$solution for the Van der Pol system \ref{['VDP']} with unprepared initial conditions \ref{['IC_nowp']} obtained at time $t=3\mu$ with $\Delta t_0 = 0.1$. The zoom of the second challenging boundary layers $t \approx 1.6\mu$ are reported. The reference solutions have been computed with the ode15s matlab solver.
• Figure 5: Implicit numerical $y-$solution for the Van der Pol system \ref{['VDP']} with unprepared initial conditions \ref{['IC_nowp']} obtained at time $t=3\mu$ with $\Delta t_0 = 0.1$. The zoom of the second challenging boundary layers $t \approx 1.6\mu$ are reported. The reference solutions have been computed with the ode15s matlab solver.