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Goal-oriented adaptivity is a powerful tool to accurately approximate physically relevant solution features for partial differential equations. In time dependent problems, we seek to represent the error in the quantity of interest as an integral over the whole space-time domain. A full space-time variational formulation allows such representation. Most authors employ implicit time marching schemes to perform goal-oriented adaptivity as it is known that they can be reinterpreted as Galerkin methods. In this work, we consider variational forms for explicit methods in time. We derive an appropriate error representation and propose a goal-oriented adaptive algorithm in space. For that, we derive the forward Euler method in time employing a discontinuous-in-time Petrov-Galerkin formulation. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy condition to ensure the stability of the method. We provide some numerical results in 1D space + time for the diffusion and advection-diffusion equations to show the performance of the proposed explicit-in-time goal-oriented adaptive algorithm.

Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit methods in time remains elusive. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous Petrov-Galerkin methods both in space and time. We build trial and test spaces for the linear diffusion equation that lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to design explicit time-domain (goal-oriented) adaptive algorithms.

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Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use 𝜑−functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the 𝜑−function action over vectors for two-dimensional (2D) matrices expressed as a Kronecker sum.For that, we present an auxiliary exponential-related matrix function that we express using Kronecker products of one-dimensional matrices. We exploit state-of-the-art implementations of 𝜑−functions to compute this auxiliary function's action and then recover the original 𝜑−action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen-Cahn equation.

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Goal-oriented adaptive algorithms produce optimal grids to solve challenging engineering problems. Recently, a novel error representation using (unconventional) pseudo-dual problems for goal-oriented adaptivity in the context of frequency-domain wave-propagation problems has been developed. In this paper, we extend this error representation to the case of time-domain problems. We express the entire problem in weak form in order to derive the adjoint formulation and apply goal-oriented adaptivity. One dimensional (1D) numerical results show that upper bounds for the new error representation are sharper than the classical ones. Therefore, this new error representation can be used to design more efficient goal-oriented adaptive methodologies.