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

中文翻译：

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显式 Runge-Kutta 方法的变分公式

在过去的几十年里，偏微分方程的变分时空公式引起了人们极大的兴趣。虽然已知隐式时间推进方案具有变分结构，但时间显式方法的伽辽金公式仍然难以捉摸。在这项工作中，我们证明了显式 Runge-Kutta 方法在空间和时间上都可以表示为不连续的 Petrov-Galerkin 方法。我们为线性扩散方程构建试验和测试空间，导致一、二和一般阶段显式 Runge-Kutta 方法。这种方法使我们能够设计显式的时域（面向目标）自适应算法