Multistage DPG time-marching scheme for nonlinear problems

TL;DR

This paper introduces high-order multistage DPG schemes for nonlinear ODEs and PDEs, extending linear methods with efficient exponential action computations and proven convergence orders.

Contribution

It develops a novel multistage DPG time-marching scheme for nonlinear systems, reducing computational cost by enabling stage post-processing without extra exponential calculations.

Findings

Methods achieve second, third, and fourth-order accuracy.

Numerical tests confirm convergence rates on semi-linear PDEs.

The approach is computationally cheaper than classical exponential Rosenbrock methods.

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 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.