The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well stablished numerical methods for solving Partial Differential Equations (PDEs) and stiff systems of Ordinary Differential Equations (ODEs), respectively. In this work, we apply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general first order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D + time linear parabolic PDEs after discretizing in space by the finite element method.

间断彼得罗夫-加勒金（DPG）方法和指数积分器是分别解决偏微分方程（PDE）和常微分方程（ODE）的刚性系统的两种成熟的数值方法。在这项工作中，我们将DPG方法应用于线性抛物线问题的时间变量，并通过分析计算出最佳测试函数。我们证明了DPG方法在时间上等效于跟踪变量的指数积分器，这些跟踪变量与内部变量解耦。此外，DPG最佳测试功能使我们能够计算时间元素内部的近似解。该DPG方法及时允许构造后验误差估计以便执行适应性。我们将这种新颖的基于DPG的时间步移方案推广到ODE的一般一阶线性系统。我们通过有限元方法在空间离散后，针对1D和2D +时间线性抛物线PDE展示了所提出方法的性能。