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间断Petrov-Galerkin（DPG）格式的构造方法，导出了适用于非线性常微分方程组的高阶多阶段DPG方法。该方法论可推广至Banach空间中的抽象发展方程，包括一类非线性偏微分方程。我们提出三种嵌套式多阶段方法：混合欧拉法、两阶段及三阶段DPG方法。类似于指数Rosenbrock方法，我们对问题进行线性化处理，因此需要计算随时间步变化的雅可比矩阵的指数作用量。构造的关键在于，其中一个阶段可通过另一阶段的后处理得到而无需额外指数步，故所提方法类比经典指数Rosenbrock方法计算成本更低。我们给出完整的收敛性证明，表明该方法分别具有二阶、三阶和四阶精度。通过对二维空间半离散化后的时间半线性偏微分方程进行时间收敛性测试，验证了所提方法的效果。