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

Abstract

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

The effect of time discretization on the solution of parabolic PDEs with ANNs

“…Moreover, the first class of methods can be profitably applied when the matrix involved in the problem has a structure that can be exploited. The introduction and study of different time-marching procedures has been proposed also for the parabolic case, for both traditional methods [16,17] and newer ones, such as Isogeometric Analysis [1,2,14] and Physics-Informed Neural Networks [9].…”