[Paper Review] Higher-order generalized-$\alpha$ methods for hyperbolic problems

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

Numerical methods for differential equations20 references4 citations

TL;DR

This paper proposes a family of higher-order generalized-$\alpha$ methods for hyperbolic problems that achieve $2k$-order accuracy in time while preserving unconditional stability and user-controlled numerical dissipation in the high-frequency range. The method extends the generalized-$\alpha$ framework by incorporating higher-order Taylor expansions and auxiliary systems, enabling efficient, stable, and dissipative control with minimal code modifications.

ABSTRACT

The generalized-$\\alpha$ time-marching method provides second-order accuracy in time and controls the numerical dissipation in the high-frequency region of the discrete spectrum. This method includes a wide range of time integrators. We increase the order of accuracy of the method while keeping the unconditional stability and the user-control on the high-frequency numerical dissipation. The dissipation is controlled by a single parameter as in the original method. Our high-order schemes require simple modifications of the available implementations of the generalized-$\\alpha$ method.

Motivation & Objective

• To develop higher-order time integration schemes for second-order hyperbolic equations that maintain the desirable properties of the generalized-$\alpha$ method.
• To overcome the second-order accuracy limitation of existing generalized-$\alpha$ methods while preserving unconditional stability and high-frequency dissipation control.
• To enable $2k$-order accuracy in time through systematic extension of the generalized-$\alpha$ formulation using higher-order Taylor terms and auxiliary systems.
• To ensure that the resulting schemes require only simple modifications to existing generalized-$\alpha$ implementations, maintaining computational efficiency.

Proposed method

• The method constructs a $k$-step formulation that solves $3k$ equations in blocks of three, with one implicit solve per block and explicit updates for the remaining variables.
• It introduces higher-order terms via Taylor expansions up to $\mathcal{O}(\tau^{3k-3})$ and defines residual corrections using auxiliary systems to enhance accuracy.
• The scheme uses a block-implicit structure where each block corresponds to a set of three equations, and the amplification matrix is analyzed to ensure stability.
• The parameters $\alpha_i$, $\gamma_i$, and $\beta_i$ are defined to control numerical dissipation, with $\rho^\infty_i$ as user-defined parameters for high-frequency damping.
• The method generalizes the original generalized-$\alpha$ method by extending its framework to $2k$-order accuracy while preserving its spectral properties.
• The resulting system is expressed in matrix form $L\mathbf{U}_{n+1} = R\mathbf{U}_n$, with the amplification matrix $L^{-1}R$ used to analyze stability and dissipation.

Experimental results

Research questions

• RQ1 Can the generalized-$\alpha$ method be extended to achieve $2k$-order accuracy in time without sacrificing unconditional stability or dissipation control?
• RQ2 How can higher-order terms be systematically incorporated into the generalized-$\alpha$ framework to improve temporal accuracy?
• RQ3 What parameterization ensures that the high-frequency numerical dissipation remains controllable while achieving higher-order accuracy?
• RQ4 Is the resulting scheme computationally efficient and backward-compatible with existing generalized-$\alpha$ implementations?
• RQ5 What is the spectral behavior of the amplification matrix for the $2k$-order scheme, and does it maintain unconditional stability?

Key findings

• The proposed method achieves $2k$-order accuracy in time for second-order hyperbolic problems, with a truncation error of $\mathcal{O}(\tau^{2k+1})$ when the parameters $\gamma_i = \alpha_i - \frac{1}{2}$ are used.
• The scheme remains unconditionally stable for all $k \geq 1$, as confirmed by spectral analysis of the amplification matrix.
• High-frequency numerical dissipation is controlled via user-defined parameters $\rho^\infty_k \in [0,1]$, with the eigenvalues of the amplification matrix approaching $\rho^\infty_k$ as the normalized frequency $\sigma \to \infty$, preserving the key feature of the original method.
• The method requires only minor modifications to existing generalized-$\alpha$ code, as the structure of the system is block-implicit and the updates are explicit for most variables.
• For $k=2$, the method achieves fourth-order accuracy, and the scheme reduces to the standard generalized-$\alpha$ method when $k=1$, confirming consistency.
• The amplification matrix structure is block-wise, with green blocks resembling the second-order generalized-$\alpha$ matrix and blue blocks being zero, enabling modular stability and dissipation analysis.