A note on the conservation properties of the generalized-$α$ method

We show that the second-order accurate generalized-$α$ 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-$α$ 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.

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