Higher-order generalized-$\alpha$ methods for parabolic problems

TL;DR

This paper introduces a new class of high-order, unconditionally stable time-marching schemes for parabolic problems, extending the generalized-$ extalpha$ method to achieve higher accuracy and controllable dissipation.

Contribution

The authors develop high-order generalized-$ extalpha$ methods that maintain stability and high-frequency dissipation control, extending the second-order approach to arbitrary order while preserving key stability properties.

Findings

Methods are A-stable and can be made L-stable with parameter adjustments.

Achieve high-order accuracy (up to (3/2k)^{th}) for even and odd k.

Maintain unconditional stability and high-frequency dissipation control.

Abstract

We propose a new class of high-order time-marching schemes with dissipation user-control and unconditional stability for parabolic equations. High-order time integrators can deliver the optimal performance of highly-accurate and robust spatial discretizations such as isogeometric analysis. The generalized-$\alpha$ method delivers unconditional stability and second-order accuracy in time and controls the numerical dissipation in the discrete spectrum's high-frequency region. Our goal is to extend the generalized-$alpha$ methodology to obtain a high-order time marching methods with high accuracy and dissipation in the discrete high-frequency range. Furthermore, we maintain the stability region of the original, second-order generalized-$alpha$ method foe the new higher-order methods. That is, we increase the accuracy of the generalized-$\alpha$ method while keeping the unconditional stability and user-control features on the high-frequency numerical dissipation. The methodology solve $k>1, k\in \mathbb{N}$ matrix problems and updates the system unknowns, which correspond to higher-order terms in Taylor expansions to obtain $(3/2k)^{th}$-order method for even $k$ and $(3/2k+1/2)^{th}$-order for odd $k$. A single parameter $\rho^\infty$ controls the dissipation, and the update procedure follows the formulation of the original second-order method. Additionally, we show that our method is A-stable and setting $\rho^\infty=0$ allows us to obtain an L-stable method. Lastly, we extend this strategy to analyze the accuracy order of a generic method.