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Multistage DPG time-marching scheme for nonlinear problems

Judit Muñoz-Matute, Leszek Demkowicz

TL;DR

This work extends the linear-time DPG framework to nonlinear evolution problems by applying a per-step linearization and constructing three nested multistage time-marching schemes: a hybrid exponential Euler method (second order), and two- and three-stage DPG methods (third and fourth order). A key feature is post-processing one internal stage to avoid an extra exponential evaluation, yielding cheaper higher-order methods compared with classical exponential Rosenbrock schemes. The authors provide stiff-order conditions, rigorous convergence proofs, and numerical tests on a 2D+time semilinear PDE, corroborating the predicted rates and showcasing energy-decay and energy-preservation properties in selected benchmarks. The results indicate that the proposed DPG-based schemes offer high accuracy with competitive computational cost for stiff nonlinear ODEs and semilinear PDEs, with potential for adaptive time stepping via nested methods and future extensions to higher-order polynomials in time.

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.

Multistage DPG time-marching scheme for nonlinear problems

TL;DR

This work extends the linear-time DPG framework to nonlinear evolution problems by applying a per-step linearization and constructing three nested multistage time-marching schemes: a hybrid exponential Euler method (second order), and two- and three-stage DPG methods (third and fourth order). A key feature is post-processing one internal stage to avoid an extra exponential evaluation, yielding cheaper higher-order methods compared with classical exponential Rosenbrock schemes. The authors provide stiff-order conditions, rigorous convergence proofs, and numerical tests on a 2D+time semilinear PDE, corroborating the predicted rates and showcasing energy-decay and energy-preservation properties in selected benchmarks. The results indicate that the proposed DPG-based schemes offer high accuracy with competitive computational cost for stiff nonlinear ODEs and semilinear PDEs, with potential for adaptive time stepping via nested methods and future extensions to higher-order polynomials in time.

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.
Paper Structure (24 sections, 5 theorems, 107 equations, 8 figures, 3 tables)

This paper contains 24 sections, 5 theorems, 107 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem, assumptions, and linearization
  3. Overview of the DPG time-marching scheme for linear problems
  4. DPG time-marching scheme for linear systems
  5. Post-processing of the trace variables
  6. Relation between the optimal test functions and the $\varphi$-functions
  7. Multistage DPG time-marching scheme for nonlinear problems
  8. Hybrid exponential Euler method
  9. Two-stage DPG method
  10. Three-stage DPG method
  11. Stiff order conditions and local truncation error
  12. Taylor expansion of the analytical solution
  13. Stiff order conditions
  14. Final methods.
  15. Convergence
  16. ...and 9 more sections

Key Result

Theorem 1

The two-stage DPG method defined in (generalDPG2) with the coefficient functions satisfying is third order, i.e.,

Figures (8)

  • Figure 1: Hybrid approximation of problem (\ref{['ODE2']}) at $I_n$.
  • Figure 2: Solution of the hybrid exponential Euler method (\ref{['ExpEuler']}).
  • Figure 3: Convergence in time at $T=1$ for the hybrid exponential Euler (\ref{['ExpEuler']}), the two-stage DPG (\ref{['generalDPG2']}), and the three-stage DPG (\ref{['generalDPG3']}) methods, and the classical exponential Euler, expbr32, and expbr32 methods.
  • Figure 4: CPU time versus number of time steps for the hybrid exponential Euler (\ref{['ExpEuler']}), the two-stage DPG (\ref{['generalDPG2']}), and the three-stage DPG (\ref{['generalDPG3']}) methods, and the classical exponential Euler, expbr32, and expbr32 methods.
  • Figure 5: Solution of the Allen-Cahn equation \ref{['AllenCahn']} obtained with the three-stage DPG method (\ref{['generalDPG3']}).
  • ...and 3 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • ...and 7 more