Stability theory of TASE-Runge-Kutta methods with inexact Jacobian

TL;DR

The paper develops a stability theory for Time-Accurate and Highly-Stable Explicit Runge-Kutta (TASE-RK) methods when the Jacobian is replaced by an inexact matrix $A$, by splitting $J=A+B$ and employing stability diagrams and the matrix field of values. It derives conditional and unconditional stability criteria first under the simplifying assumption of simultaneous diagonalizability, then in the general case using field-of-values arguments, and finally demonstrates the approach on Keller–Segel/Fisher–Kolmogorov-type PDEs and benchmark PDEs like Burgers and FitzHugh–Nagumo, showing that carefully chosen $A$ yields strong stability and computational savings relative to Rosenbrock-like methods. The results include explicit step-size bounds for $p=2,3,4$ and practical guidelines for applying the theory to nonlinear problems, with numerical experiments confirming robustness and efficiency. The work provides a foundational framework for assessing and ensuring stability of W-method subclasses with inexact Jacobians and motivates extensions to nonlinear stability and broader W-methods.

Abstract

This paper analyzes the stability of the class of Time-Accurate and Highly-Stable Explicit Runge-Kutta (TASE-RK) methods, introduced in 2021 by Bassenne et al. (J. Comput. Phys.) for the numerical solution of stiff Initial Value Problems (IVPs). Such numerical methods are easy to implement and require the solution of a limited number of linear systems per step, whose coefficient matrices involve the exact Jacobian $J$ of the problem. To significantly reduce the computational cost of TASE-RK methods without altering their consistency properties, it is possible to replace $J$ with a matrix $A$ (not necessarily tied to $J$) in their formulation, for instance fixed for a certain number of consecutive steps or even constant. However, the stability properties of TASE-RK methods strongly depend on this choice, and so far have been studied assuming $A=J$. In this manuscript, we theoretically investigate the conditional and unconditional stability of TASE-RK methods by considering arbitrary $A$. To this end, we first split the Jacobian as $J=A+B$. Then, through the use of stability diagrams and their connections with the field of values, we analyze both the case in which $A$ and $B$ are simultaneously diagonalizable and not. Numerical experiments, conducted on Partial Differential Equations (PDEs) arising from applications, show the correctness and utility of the theoretical results derived in the paper, as well as the good stability and efficiency of TASE-RK methods when $A$ is suitably chosen.

Figures (9)

• Figure 1: Function $\hat{T}_p(y)(=yT_p(y))$, for $p=2,3,4$. For each $p$, $\hat{T}_p$ is an increasing function whose values are all within the interval $[\hat{T}_p^*,0]$, where $\hat{T}_2^*\approx -1$, $\hat{T}_3^*\approx -1.5961$, $\hat{T}_4^*\approx -1.5961$.
• Figure 1: Stability diagrams $\mathcal{D}_{y_i,p}$ of TRK2 (left) and TRK3 (right) methods, for several values of $y_i=k\lambda_i$, i.e. $(\lambda_1,\lambda_2,\lambda_3)=(-100,-10,-1)$, $k=7.8390e-01$ for TRK2, $k=2.8428e-01$ for TRK3; the points plotted inside the figures correspond to $(\mu_1,\mu_2,\mu_3)=(\frac{1}{2},\frac{6}{5},\frac{3}{2})$.
• Figure 1: $\mathcal{W}_{1}(-A,B)$ (left and right), and $-\mathcal{D}_{y,2}$, $y=-2.1$ (left), $-\mathcal{D}_{y,3}$, $y=-1.45$ (right), with $A$ and $B$ as in \ref{['ex2m']}.
• Figure 1: Numerical solution of the semi-discretized FK equation by the TRK3 method, with $k=0.5$, setting parameters, initial and boundary conditions as explained in Subsection \ref{['subsecFK']}.
• Figure 1: $\mathcal{W}_{1}(-A,B)$, with $A$ as in \ref{['BurgA']}, $\epsilon=10^{-1}, \ \kappa=1$, and $-\mathcal{ D}_{\infty,p}$ for $p=2,3,4$ (top left); solution of Burgers' equation by TRK2 method, with $k=0.5, \ \kappa=1$, and $\epsilon=10^{-1}$ (top right), $\epsilon=10^{-2}$ (bottom right); comparison in terms of CPU time and error between ROS and TRK methods on Burgers' equation, $\epsilon=10^{-1}$ (bottom left).

Theorems & Definitions (34)

• Remark 2.1
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Lemma 3.4
• Proof 1
• Corollary 3.5
• Proof 2
• Theorem 3.6
• Proof 3