Domain Overlapping Algorithm with Nonlinear Mapping for Collocation-Based Solutions of Eigenvalue Problems

TL;DR

This paper introduces four innovative domain decomposition algorithms with nonlinear mapping to improve collocation-based eigenvalue problem solutions, achieving higher accuracy and efficiency near discontinuities and steep gradients.

Contribution

The paper presents novel algorithms that enhance spectral accuracy and node clustering flexibility for eigenvalue problems with sharp interfaces, surpassing existing methods like Chebfun.

Findings

One-point overlap method reduces grid points significantly while maintaining spectral convergence.

Two-point overlap method allows arbitrary node distributions with exponential error reduction.

Validated on 3D flow and Burgers equation, matching prior results with fewer nodes.

Abstract

This paper presents four novel domain decomposition algorithms integrated with nonlinear mapping techniques to address collocation-based solutions of eigenvalue problems involving sharp interfaces or steep gradients. The proposed methods leverage the spectral accuracy of Chebyshev polynomials while overcoming limitations of existing tools like Chebfun, particularly in preserving higher-order derivative continuity and enabling flexible node clustering near discontinuities. Key findings include the following: for algorithm Performance: The one-point overlap method demonstrated significant improvements over global mapping approaches, reducing required grid points by orders of magnitude while maintaining spectral convergence. The two-point overlap method further methods generalized the approach, allowing arbitrary node distributions and nonlinear mappings. These achieved exponential error reduction for Burgers equation) by combining Taylor expansions with Chebyshev derivatives in overlap regions. While Chebfun splitting strategy automates domain decomposition, it enforces only C0 continuity, leading to discontinuous higher derivatives. In contrast, the proposed algorithms preserved smoothness up to CN continuity, critical for eigenvalue problems in hydrodynamic stability and nonlinear BVPs. Validation on 3D channel flow with viscosity stratification and Burgers equation highlighted the methods robustness. For instance, eigenvalue calculations for miscible core-annular flows matched prior results while resolving sharp viscosity gradients with fewer nodes.