[Paper Review] High-order generalized-$\alpha$ methods

Quanling Deng, Pouria Behnoudfar+1 more|arXiv (Cornell University)|Feb 14, 2019

Numerical methods for differential equations8 references4 citations

TL;DR

This paper proposes a third-order accurate generalized-$\alpha$ time integrator that maintains unconditional stability and user-controlled high-frequency dissipation, extending the second-order generalized-$\alpha$ method via higher-order Taylor expansions in the discrete formulation. The method achieves third-order accuracy while preserving the unconditionally stable, single-step structure of the original scheme, with numerical dissipation controlled through parameters $\alpha_m$ and $\alpha_f$.

ABSTRACT

The generalized-$\\alpha$ method encompasses a wide range of time integrators. The method possesses high-frequency dissipation while minimizing unwanted low-frequency dissipation and the numerical dissipation can be controlled by the user. The method is unconditionally stable and is of second-order accuracy in time. We extend the second-order generalized-$\\alpha$ method to third-order in time while the numerical dissipation can be controlled in a similar fashion. We establish that the third-order method is unconditionally stable. We discuss a possible path to the generalization to higher order schemes. All these high-order schemes can be easily implemented into programs that already contain the second-order generalized-$\\alpha$ method.

Motivation & Objective

• To extend the second-order generalized-$\alpha$ method to third-order accuracy in time while preserving its key features.
• To maintain unconditional stability in the new high-order scheme.
• To retain user control over high-frequency numerical dissipation, a hallmark of the generalized-$\alpha$ method.
• To enable seamless integration into existing second-order generalized-$\alpha$ codebases.
• To provide a framework for generalizing to even higher-order schemes.

Proposed method

• Reformulate the generalized-$\alpha$ method by treating $U_{n}^{\alpha_f}$ and $V_{n}^{\alpha_m}$ as higher-order Taylor expansions rather than linear sub-steps.
• Introduce additional parameters $\alpha_f$ and $\alpha_m$ to represent higher-order terms in the Taylor expansion of $U_{n+\alpha_f}$ and $V_{n+\alpha_m}$, enabling third-order accuracy.
• Derive the time-marching scheme using Taylor series analysis to determine free parameters $\alpha_m$ and $\alpha_f$ for optimal accuracy.
• Construct the amplification matrix and analyze its eigenvalues to establish the unconditional stability region.
• Control numerical dissipation by setting the high-frequency eigenvalue $\rho_\infty$ to a user-defined value in $[0,1]$.
• Use symbolic and numerical analysis to verify unconditional stability and determine the parameter region for stability and desired dissipation.

Experimental results

Research questions

• RQ1 Can the generalized-$\alpha$ method be extended to third-order accuracy in time without sacrificing unconditional stability?
• RQ2 How can user-controlled high-frequency dissipation be preserved in a higher-order time integrator?
• RQ3 What parameter choices $\alpha_m$ and $\alpha_f$ yield third-order accuracy and unconditional stability?
• RQ4 Can the resulting scheme be implemented with minimal changes to existing second-order generalized-$\alpha$ code?
• RQ5 What is the stability region of the third-order generalized-$\alpha$ method, and how does it compare to the second-order version?

Key findings

• The proposed third-order generalized-$\alpha$ method achieves formal third-order accuracy in time through higher-order Taylor expansion of the solution and velocity terms.
• The method is unconditionally stable for all $\alpha_m \geq \frac{7}{12}$ and $\frac{1}{2} \leq \alpha_f \leq \alpha_m - \frac{1}{12}$, with the stability region confirmed symbolically and numerically.
• Numerical dissipation at high frequencies is controlled by the parameter $\rho_\infty$, with $\alpha_m = \frac{13 + 20\rho_\infty - 5\rho_\infty^2}{12(\rho_\infty + 1)^2}$ and $\alpha_f = \frac{1 + 3\rho_\infty}{2(\rho_\infty + 1)^2}$ for $0 \leq \rho_\infty \leq 1$.
• The amplification matrix eigenvalues are bounded by 1 in magnitude across the unconditional stability region, confirming stability.
• The third-order scheme exhibits a larger unconditional stability region than the second-order version, as shown in numerical plots.
• The method remains a single-step, implicit scheme that can be easily integrated into existing second-order generalized-$\alpha$ solvers with minimal code changes.