[Paper Review] Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains

Mingtao Xia, Sihong Shao+1 more|arXiv (Cornell University)|Sep 28, 2020

Ocean Waves and Remote Sensing23 references6 citations

TL;DR

This paper proposes adaptive scaling and moving techniques for spectral methods in unbounded domains using frequency and exterior-error indicators to dynamically adjust collocation points. The methods enable fast spectral convergence for time-dependent problems involving diffusion and translation, with demonstrated success in tracking blowup in cell proliferation models and outperforming fixed-scaling approaches.

ABSTRACT

When using Laguerre and Hermite spectral methods to numerically solve PDEs in unbounded domains, the number of collocation points assigned inside the region of interest is often insufficient, particularly when the region is expanded or translated to safely capture the unknown solution. Simply increasing the number of collocation points cannot ensure a fast convergence to spectral accuracy. In this paper, we propose a scaling technique and a moving technique to adaptively cluster enough collocation points in a region of interest in order to achieve a fast spectral convergence. Our scaling algorithm employs an indicator in the frequency domain that is used to determine when scaling is needed and informs the tuning of a scaling factor to redistribute collocation points to adapt to the diffusive behavior of the solution. Our moving technique adopts an exterior-error indicator and moves the collocation points to capture the translation. Both frequency and exterior-error indicators are defined using only the numerical solutions. We apply our methods to a number of different models, including diffusive and moving Fermi-Dirac distributions and nonlinear Dirac solitary waves, and demonstrate recovery of spectral convergence for time-dependent simulations. Performance comparison in solving a linear parabolic problem shows that our frequency scaling algorithm outperforms the existing scaling approaches. We also show our frequency scaling technique is able to track the blowup of average cell sizes in a model for cell proliferation.

Motivation & Objective

• Address the challenge of insufficient collocation points in regions of interest when solving PDEs in unbounded domains using spectral methods.
• Overcome the limitations of fixed scaling and moving strategies that fail to adapt to evolving solution behavior.
• Develop indicators based solely on numerical solutions to enable automatic, adaptive redistribution of collocation points without prior knowledge of the solution.
• Ensure spectral convergence for time-dependent problems involving both diffusion and translation by combining scaling and moving techniques.
• Demonstrate the effectiveness of the proposed methods on nonlinear and blowup problems, including cell proliferation models.

Proposed method

• Introduce a frequency indicator derived from the Fourier transform of intermediate numerical solutions to detect high-frequency components and assess approximation error.
• Use the frequency indicator to guide time-dependent scaling of the coordinate system, adjusting the scaling factor β(t) to maintain low error and preserve spectral convergence.
• Propose an exterior-error indicator to estimate error in the outer domain, enabling adaptive movement of collocation points to follow translating solutions.
• Implement a 'first moving then scaling' strategy that combines both indicators to handle problems with both translation and diffusion.
• Apply the methods within a pseudo-spectral framework using generalized Laguerre and Hermite functions for spatial discretization.
• Use only numerical solutions to compute indicators, avoiding reliance on analytical knowledge of the solution or its derivatives.

Experimental results

Research questions

• RQ1 How can spectral methods in unbounded domains be made adaptive to evolving solution features such as diffusion and translation?
• RQ2 Can frequency-domain indicators derived from numerical solutions be used to guide dynamic scaling for improved spectral convergence?
• RQ3 How can the error in the outer domain be estimated and used to trigger adaptive mesh movement in unbounded domains?
• RQ4 To what extent can adaptive scaling and moving techniques outperform fixed-scaling approaches in time-dependent PDEs?
• RQ5 Can the proposed methods accurately track blowup behavior in unbounded domains, such as in cell proliferation models?

Key findings

• The frequency-dependent scaling algorithm achieves faster spectral convergence than fixed-scaling methods in solving linear parabolic problems, with error reduced from 1.143e-02 to below 8.662e-06 at t=10 when N=20.
• The frequency indicator is maintained at approximately 10^-6 during time evolution by dynamically adjusting the scaling factor βx from 0.9 to 0.2766, ensuring high accuracy.
• The average size of the solution in the cell proliferation model is accurately tracked, with the scaled solution matching the analytical mean size ⟨x(t)⟩=5+t to within 15.001 at t=10.
• The method successfully captures blowup behavior in a structured cell population model, where the average size grows linearly over time and requires adaptive scaling to remain resolved.
• The combination of moving and scaling techniques enables robust resolution of problems with both translation and diffusion, maintaining spectral convergence without prior knowledge of solution dynamics.
• The exterior-error indicator effectively detects when solution translation requires mesh movement, allowing the method to adaptively shift collocation points to the region of interest.