A note on the conservation properties of the generalized-$\alpha$ method

TL;DR

This paper reveals that the generalized-$\alpha$ method can be interpreted as an implicit midpoint method on a shifted mesh, ensuring conservation properties when combined with conservative spatial discretizations and uniform time steps.

Contribution

It provides a novel interpretation of the generalized-$\alpha$ method as an implicit midpoint method on a shifted mesh, demonstrating its conservation properties under specific conditions.

Findings

The generalized-$\alpha$ method is equivalent to an implicit midpoint method on a shifted mesh.

The fully-discrete method preserves discrete balance laws under certain conditions.

Conservation properties are guaranteed when using second-order accuracy, uniform meshes, and conservative spatial discretizations.

Abstract

We show that the second-order accurate generalized-$\alpha$ method on a uniform temporal mesh may be viewed as an implicit midpoint method on a shifted temporal mesh. With this insight, we demonstrate generalized-$\alpha$ time integration of a finite element spatial discretization of a conservation law system results in a fully-discrete method admitting discrete balance laws when (i) the time integration is second-order accurate, (ii) a uniform temporal mesh is employed, (iii) the spatial discretization is conservative, and (iv) conservation variables are discretized.