Multistage Discontinuous Petrov–Galerkin Time-Marching Scheme for Nonlinear Problems

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1956-1978, August 2024.
Abstract. 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 nonlinear 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 step to time step. The key point of our construction is that one of the stages can be postprocessed 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 semilinear partial differential equation after a semidiscretization in space.