A New Class of General Linear Method with Inherent Quadratic Stability for Solving Stiff Differential Systems

This article proposes a new class of general linear method with $p=q$ and $r=s=p+1$. The construction of the present method is carried out using order conditions and error minimization subject to $A$- stability constraints. The proposed time integration schemes are $A$- and $L$-stable general linear methods (GLMs) equipped with inherent quadratic stability (IQS) criteria. We construct implicit GLMs of orders up to four with $p = q$ and $s = r$ along with the Nordsieck input vector assumption. Further, we test these schemes on three real-world problems: the van der Pol oscillator and two partial differential equations consisting of diffusion (Burgers’ equation and the Gray-Scott model), and numerical results are presented. Computational results confirm that our proposed schemes are competitive with the existing GLMs and can be recognized as an alternative time integration scheme. We demonstrate the order of accuracy and convergence for the proposed schemes through observed order computation and error versus step size plots.

💡 Research Summary

The paper introduces a novel class of implicit general linear methods (GLMs) specifically designed for stiff differential systems. The authors focus on methods where the order of accuracy p equals the stage order q, and the number of internal and external stages satisfies r = s = p + 1. This configuration enables the use of a Nordsieck input vector, which stores the solution and its scaled derivatives, facilitating variable step‑size and variable‑order implementations.

The construction proceeds by first writing the GLM in the standard form with coefficient matrices A, U, B, and V. The matrices A and V are given a special lower‑triangular and unit‑row structure, respectively, with a positive parameter λ controlling the implicitness. Order conditions are derived from Theorem 2.2, ensuring that the method attains order p and stage order q. These conditions are expressed as linear relations among the coefficients and the Nordsieck basis vectors.

A central contribution is the enforcement of inherent quadratic stability (IQS). IQS is defined through matrix equivalence relations BA ≡ XB and BU ≡ XV − VX, where X has non‑zero entries only in its first two rows. By satisfying these relations, the stability polynomial of the method collapses to a quadratic factor multiplied by η^{r‑2}, i.e.,

p(η, ω) = η^{r‑2}