In this article, we employ the construction of the time-marching Discontinuous Petrov-Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for non-linear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two- and three-stage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time steps. The key point of our construction is that one of the stages can be post-processed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second, third, and fourth-order accurate, respectively. We test the convergence in time of our methods on a 2D + time semi-linear partial differential equation after a semidiscretization in space.

翻译：本文采用我们为线性问题开发的时间推进不连续Petrov-Galerkin（DPG）格式的构造方法，推导出适用于非线性常微分方程组的高阶多阶段DPG方法。该方法可推广至巴拿赫空间中的抽象发展方程，包括一类非线性偏微分方程。我们提出了三种嵌套的多阶段方法：混合欧拉方法、两阶段和三阶段DPG方法。与指数型Rosenbrock方法类似，我们采用问题的线性化处理，因此需要计算随时间步变化的雅可比矩阵的指数作用。我们构造的关键在于：其中一个阶段可通过另一阶段的后处理获得，无需额外指数步。因此，我们提出的方法类别在计算上比经典的指数型Rosenbrock方法更经济。我们给出了完整的收敛性证明，表明这些方法分别具有二阶、三阶和四阶精度。我们通过一个二维+时间的半线性偏微分方程在空间半离散化后的数值实验，验证了所提方法的时间收敛性。