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

TL;DR

Problem: eigenvalue computations for hydrodynamic stability and nonlinear boundary-value problems are difficult when sharp interfaces or steep gradients are present in finite domains. Approach: four domain-overlapping algorithms couple nonlinear mappings with Chebyshev spectral collocation to cluster points near interfaces while preserving smoothness up to $C^N$; overlaps range from one point to multi-point overlaps across multi-interval domains, with Taylor expansions linking subdomains. Contributions: the one-point method reduces required nodes from about 6500 in a global mapping to a few hundred while preserving spectral convergence; the two-point and multi-point methods support arbitrary node distributions and achieve exponential error reduction for the Burgers equation via overlap-based Taylor-Chebyshev coupling; the methods preserve higher derivatives ($C^N$) unlike Chebfun's $C^0$-split representations and succeed in challenging test cases like miscible core-annular flows and 3D channel flow with viscosity stratification. Significance: the framework delivers high-accuracy, memory-efficient eigenvalue and BVP solutions on complex multi-interval domains, enabling robust hydrodynamic stability analyses with sharp gradients.

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.

Figures (14)

• Figure 1: Non-zero matrix elements distribution of the semi-global method: (1) The first order derivative matrix and (b) the second order derivative matrix.
• Figure 2: Comparison of the error in approximating the function for a solution using chebfun with the results of the one-point overlap method.
• Figure 3: Comparison of the error in approximating the function for solution using chebfun with the results of the one-point overlap method.
• Figure 4: 3 D channel flow instability with $\delta = 0.02$: (a) velocity profile and its derivative (b) Spectral distribution when wave number $k=1-i$
• Figure 5: (a) Comparison of our calculations with the core-annular pipe flow calculations in Fig.3(a) of Selvam2009 for the axisymmetric mode. The abscissa $r_0$ corresponds to various locations of the diffusive interface of width $\delta$, for the parameter matrix $(M,Re,Sc,\delta^*)=(25,48,7500,[0.02,0.03])$. (b) Comparison with the results from Selvam in Fig.7(2007), considering both helical and axisymmetric modes when the ratio of annular to core fluid viscosity is 2.718.