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

TL;DR

This paper develops higher-order generalized-alpha methods for hyperbolic problems that maintain stability and dissipation control, requiring minimal modifications to existing implementations.

Contribution

It introduces higher-order accurate versions of the generalized-alpha method that preserve stability and dissipation control with simple implementation changes.

Findings

Achieves higher-order accuracy while maintaining unconditional stability.

Preserves user-control over high-frequency numerical dissipation.

Requires minimal modifications to existing generalized-alpha implementations.

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.