Variational formulations for explicit Runge-Kutta Methods

摘要: Abstract Variational space-time formulations for partial differential equations have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes variational structure, are often employed adaptivity. Previously, Galerkin explicit methods were introduced ordinary employing specific inexact quadrature rules. In this work, we prove Runge-Kutta can be expressed as discontinuous-in-time Petrov-Galerkin linear diffusion equation. We systematically build trial and test functions that, after exact integration time, lead one, two, general stage methods. This approach enables us reproduce existing time-domain (goal-oriented) adaptive algorithms using time.