Efficient Nonlinear Waveform Relaxation Method for Solving Large Systems of Ordinary Differential Equations

Core Concepts

The paper presents a nonlinear waveform relaxation method for efficiently solving large systems of nonlinear ordinary differential equations. The method employs an inner-outer iterative approach, where the inner iterations use a block Krylov subspace method to solve the linear subproblems arising from the waveform relaxation.

Abstract

The paper considers the problem of efficiently integrating large systems of nonlinear ordinary differential equations in time. It presents 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.
The key aspects of the approach are:

The linear subproblems are solved using the exponential block Krylov subspace (EBK) method, which allows for an across-time parallelism and can be seen as an efficient version of the time-parallel PARAEXP method.

Theoretical convergence analysis is provided for the nonlinear waveform relaxation method, showing that the iteration converges under certain conditions on the Lipschitz constant and the time interval length.

An error estimate is derived in terms of a computable nonlinear residual, allowing the convergence to be monitored in practice.

The impact of inexactly solving the linear subproblems on the nonlinear iteration convergence is analyzed.

Numerical experiments are conducted on three nonlinear test problems: the 1D Burgers equation, the 3D Liouville-Bratu-Gelfand equation, and a 3D nonlinear heat conduction problem. The results demonstrate the efficiency of the proposed method compared to conventional time-stepping integrators.

Stats

The paper presents several numerical results comparing the performance of the nonlinear waveform relaxation method with the MATLAB ode15s solver. Key data points include:

For the 1D Burgers equation with viscosity ν = 3 × 10^-4, the nonlinear EBK method required 5-11 nonlinear iterations and 5-11 LU factorizations to achieve a relative error of around 5 × 10^-6 to 5 × 10^-5 for final times T = 0.5, 1.0, 1.5. The ode15s solver required 59-103 time steps and 14-23 LU factorizations to achieve a similar accuracy.
For the 1D Burgers equation with viscosity ν = 3 × 10^-5, the results were similar, with the nonlinear EBK method being more efficient than ode15s.
The number of nonlinear iterations in the EBK method remained practically constant as the grid size was increased, while the number of LU factorizations in ode15s increased with the grid size.

Quotes

"The key attractive feature of the waveform relaxation methods is that they employ this linear algebra machinery across a certain time interval rather than within a time step, so that computational costs are distributed across time. This leads to a higher computational efficiency as well as to a parallelism across time."
"Convergence of the waveform relaxation methods has been studied in different settings, in particular, for linear initial-value problems, for nonlinear Gauss-Seidel and Jacobi iterations (the working horses of classical waveform relaxation) and for time-discretized settings, see [22, 23] for a survey. Convergence results for waveform relaxation in general nonlinear settings are scarce, and it is often assumed that 'studying the linear case carefully . . . might be what users really need' [23]."

Key Insights Distilled From

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

by Mike A. Botc... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2307.00276.pdf

Deeper Inquiries

How can the nonlinear waveform relaxation method be extended to handle more complex nonlinear systems, such as those arising in fluid dynamics or structural mechanics

The nonlinear waveform relaxation method can be extended to handle more complex nonlinear systems, such as those arising in fluid dynamics or structural mechanics, by incorporating advanced techniques and algorithms. One approach is to enhance the efficiency and accuracy of the method by utilizing higher-order numerical schemes for spatial discretization. This can help in capturing intricate flow patterns or structural responses more effectively. Additionally, the method can be adapted to handle multi-physics problems by incorporating coupling between different physical phenomena, such as fluid-structure interaction or heat transfer in solid structures.
Furthermore, the nonlinear waveform relaxation method can be extended to handle adaptive mesh refinement strategies. By dynamically adjusting the spatial grid resolution based on the solution behavior, the method can effectively capture localized features or discontinuities in the system. This adaptive approach can improve the overall accuracy and efficiency of the method, especially in scenarios where the solution exhibits varying scales of behavior.
In the context of fluid dynamics, the method can be enhanced to handle turbulent flows by incorporating turbulence models or large eddy simulation techniques. These additions can improve the representation of turbulent structures and enhance the predictive capabilities of the method for complex flow scenarios. Similarly, in structural mechanics, the method can be extended to handle nonlinear material behavior, contact mechanics, or dynamic loading conditions to accurately simulate the response of complex structures under varying conditions.

What are the potential limitations of the method, and how could it be further improved to handle a wider range of problems

While the nonlinear waveform relaxation method offers several advantages in terms of computational efficiency and parallelism across time, there are potential limitations that need to be addressed for handling a wider range of problems. One limitation is the convergence behavior of the method, especially for highly nonlinear systems or stiff differential equations. Improving the convergence properties through advanced iterative solvers, adaptive strategies, or preconditioning techniques can enhance the robustness of the method for a broader class of problems.
Another limitation is the scalability of the method for large-scale simulations. As the size of the system increases, the computational cost of solving the linear systems at each iteration can become prohibitive. To address this limitation, parallelization strategies, domain decomposition methods, or hybrid solvers can be employed to distribute the computational workload and improve the scalability of the method for high-performance computing environments.
To further improve the method, incorporating higher-order temporal discretization schemes, such as implicit-explicit methods or multi-step methods, can enhance the accuracy and stability of the solution, especially for stiff or oscillatory systems. Additionally, exploring adaptive time-stepping strategies or error estimation techniques can help in dynamically adjusting the time integration process to efficiently capture the dynamics of the system.

What other types of time-parallel or across-time integration methods could be combined with the nonlinear waveform relaxation approach to achieve even greater computational efficiency

The nonlinear waveform relaxation approach can be combined with other time-parallel or across-time integration methods to achieve even greater computational efficiency and accuracy. One potential approach is to integrate the method with spectral methods, such as Fourier or Chebyshev spectral collocation methods, to efficiently handle problems with smooth solutions or periodic behavior. By leveraging the spectral accuracy of these methods, the overall convergence and efficiency of the waveform relaxation approach can be enhanced.
Another strategy is to combine the nonlinear waveform relaxation method with reduced-order modeling techniques, such as proper orthogonal decomposition (POD) or balanced truncation, to reduce the dimensionality of the system and accelerate the solution process. By capturing the dominant modes of the system dynamics, these model reduction techniques can significantly reduce the computational cost while maintaining the accuracy of the solution.
Furthermore, integrating the nonlinear waveform relaxation method with machine learning algorithms, such as neural networks or reinforcement learning, can offer new opportunities for adaptive and data-driven time integration strategies. By leveraging the predictive capabilities of machine learning models, the method can adaptively adjust the time integration process based on the system behavior, leading to improved efficiency and accuracy in solving complex nonlinear systems.