Split generalized-$\alpha$ method: A linear-cost solver for a modified generalized-method for multi-dimensional second-order hyperbolic systems

TL;DR

This paper introduces a linear-cost splitting technique for the generalized-$ extalpha$ method, enabling efficient and stable solutions of multi-dimensional second-order hyperbolic PDEs using finite elements or isogeometric analysis.

Contribution

It develops a variational splitting approach with linear computational complexity for multi-dimensional hyperbolic systems, ensuring unconditional stability and optimal convergence.

Findings

Linear computational cost growth with degrees of freedom

Unconditional stability established through spectral analysis

Optimal convergence in space and second-order accuracy in time

Abstract

We propose a variational splitting technique for the generalized-$\alpha$ method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows linearly with respect to the total number of degrees of freedom for multi-dimensional problems. We consider standard $C^0$ finite elements as well as smoother B-splines in isogeometric analysis for the spatial discretization. We also study the spectrum of the amplification matrix to establish the unconditional stability of the method. We then show that the stability behavior affects the overall behavior of the integrator on the entire interval and not only at the limits $0$ and $\infty$. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. For the numerical tests, we compute the $L_2$ and $H^1$ norms to show the optimal convergence of the discrete method in space and second-order accuracy in time.