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On convergence of waveform relaxation for nonlinear systems of ordinary differential equations

Mike A. Botchev

TL;DR

This work analyzes a nonlinear waveform relaxation scheme where each outer iteration solves a linear IVP with an exponential block Krylov (EBK) method, enabling time-parallel simulation within PARAEXP. Convergence is established under a splitting Φ(t,y) = -A_k y + f_k(y) + g(t), with Lipschitz constant L, a bound on the linear part via ω, and a time window length T, yielding a contraction condition C L T φ_1(-ω T) < 1 and a proven superlinear rate. The residual of the nonlinear problem is linked to the actual error, and the impact of inexact inner solves is quantified, showing that controlled linear residuals preserve convergence. Numerical experiments on 1D Burgers, 3D Liouville-Bratu-Gelfand, and 3D nonlinear heat conduction demonstrate that the nonlinear EBK approach can outperform traditional implicit time-stepping (ode15s) and ROS2 in many scenarios, particularly when the linear solves are efficiently managed via shift-and-invert EBK, though the method has limitations for strongly nonlinear dynamics and requires tuning. Overall, the work provides a rigorous and practical framework for a grid- and time-interval-independent, time-parallel nonlinear solver with clear guidance on convergence and implementation trade-offs.

Abstract

To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard-Lindelöf iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, three-dimensional Liouville-Bratu-Gelfand, and three-dimensional nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators.

On convergence of waveform relaxation for nonlinear systems of ordinary differential equations

TL;DR

This work analyzes a nonlinear waveform relaxation scheme where each outer iteration solves a linear IVP with an exponential block Krylov (EBK) method, enabling time-parallel simulation within PARAEXP. Convergence is established under a splitting Φ(t,y) = -A_k y + f_k(y) + g(t), with Lipschitz constant L, a bound on the linear part via ω, and a time window length T, yielding a contraction condition C L T φ_1(-ω T) < 1 and a proven superlinear rate. The residual of the nonlinear problem is linked to the actual error, and the impact of inexact inner solves is quantified, showing that controlled linear residuals preserve convergence. Numerical experiments on 1D Burgers, 3D Liouville-Bratu-Gelfand, and 3D nonlinear heat conduction demonstrate that the nonlinear EBK approach can outperform traditional implicit time-stepping (ode15s) and ROS2 in many scenarios, particularly when the linear solves are efficiently managed via shift-and-invert EBK, though the method has limitations for strongly nonlinear dynamics and requires tuning. Overall, the work provides a rigorous and practical framework for a grid- and time-interval-independent, time-parallel nonlinear solver with clear guidance on convergence and implementation trade-offs.

Abstract

To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard-Lindelöf iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, three-dimensional Liouville-Bratu-Gelfand, and three-dimensional nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators.
Paper Structure (11 sections, 5 theorems, 71 equations, 4 figures, 3 tables)

This paper contains 11 sections, 5 theorems, 71 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Nonlinear waveform relaxation and its convergence
  3. Problem setting, assumptions and notation
  4. Nonlinear waveform relaxation
  5. Residual of the nonlinear problem
  6. Linear inner iteration and its residual
  7. Implementation of nonlinear waveform relaxation
  8. Numerical experiments
  9. 1D Burgers equation
  10. 3D Bratu test problem
  11. Conclusions

Key Result

Proposition 1

Let IVP IVP0 be solved iteratively by nonlR and let assumptions L,A hold for $A_k$ and $f_k$ in nonlR, $k=0,1,\dots$. Then for the error $\epsilon_{k+1}(t)\equiv y(t) - y_{k+1}(t)$ of the iterative approximation $y_{k+1}(t)$ holds, for $k=0,1,\dots$, and nonlinear waveform relaxation nonlR converges for $t\in[0,T]$ to solution $y(t)$ of IVP IVP0,split_f provided that

Figures (4)

  • Figure 1: The Burgers test. Number of required LU factorizations versus the grid size in the nonlinear EBK solver (left) and ode15s (right) for different final time values $T$ and viscosity $\nu=3\cdot 10^{-4}$
  • Figure 2: The Burgers test. The first three iterative approximations $y_0(T)=v$, $y_1(T)$, $y_2(T)$ and reference solution $y_{\mathrm{ref}}(T)$ for the Burgers test, grid size $n=2000$, final time $T=1.5$, viscosity $\nu=3\cdot 10^{-4}$.
  • Figure 3: The Burgers test. Residual norm \ref{['rk']} and relative error norm \ref{['err_reached']} versus iteration number for $T=0.5$ (top plots) and $T=1.5$ (bottom plots), $\nu=3\cdot 10^{-4}$ (left plots) and $\nu=3\cdot 10^{-5}$ (right plots). The grid size is $n=4000$. Error stagnation visible at the bottom right plot can be repaired by slightly increasing the block size $m$.
  • Figure 4: Left: initial value function $u(x,y,z,0)$ of the Bratu test on a uniform $40\times 40\times 40$ grid for $z=0.5$. Right: numerical solution $u(x,y,z,t)$ on the same grid for $z=0.5$ and $t=1\cdot 10^{-3}$.

Theorems & Definitions (13)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 3 more