# High-order generalized-$\alpha$ methods

## TL;DR

This paper extends the generalized-$ extalpha$ method to third-order accuracy, maintaining stability and controllable numerical dissipation, thus enhancing time integration techniques for computational simulations.

## Contribution

The paper introduces a third-order generalized-$ extalpha$ method, proving its unconditional stability and discussing potential for higher-order generalizations.

## Key findings

- Third-order method is unconditionally stable.
- Numerical dissipation can be controlled similarly to second-order.
- Easy integration into existing second-order generalized-$ extalpha$ frameworks.

## 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.