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

TL;DR

This paper introduces a new class of general linear methods with inherent quadratic stability designed for solving stiff differential systems, demonstrating their effectiveness through theoretical analysis and real-world problem testing.

Contribution

The paper develops a novel class of $A$- and $L$-stable general linear methods with inherent quadratic stability, constructed using order conditions and error minimization.

Findings

Methods are $A$- and $L$-stable with inherent quadratic stability.

Constructed implicit GLMs of orders up to four.

Numerical results show competitiveness with existing schemes.

Abstract

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.