An Adaptive Orthogonal Basis Method for Computing Multiple Solutions of Differential Equations with polynomial nonlinearities

TL;DR

The paper tackles the challenge of computing multiple solutions to nonlinear differential equations with polynomial nonlinearities. It introduces an Adaptive Orthogonal Basis Method that dynamically builds a reduced orthogonal basis and uses companion-matrix information to generate diverse initial guesses, all within a spectral Chebyshev-Collocation framework and solved by a robust trust-region method. The approach avoids predefined basis pools, demonstrates high accuracy and efficiency in 1D and 2D experiments, and shows potential for broad applicability to nonlinear PDEs. This method offers a scalable, robust avenue for uncovering multiple states in nonlinear systems where traditional methods may struggle or be prohibitively expensive.

Abstract

This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of predefining candidate basis pools, our novel method adaptively computes bases, considering the equation's nature and structural characteristics of the solution. It further leverages companion matrix techniques to generate initial guesses for subsequent computations. Thus this approach not only yields numerous initial guesses for solving such equations but also adapts orthogonal basis functions to effectively address discretized nonlinear systems. Through a series of numerical experiments, this paper demonstrates the method's effectiveness and robustness. By reducing computational costs in various applications, this novel approach opens new avenues for uncovering multiple solutions to differential equations with polynomial nonlinearities.

Figures (17)

• Figure 4.1: The flow chart of the adaptive orthogonal basis method for computing multiple solutions of \ref{['eq4.1']}. Two basis functions, $\phi_{0}$ and $\phi_{1}$, form two solutions, $\hat{u}^{(0)}_{1}$ and $\hat{u}^{(1)}_{1}$, respectively, in Step 4. A notable difference between $\hat{u}^{(0)}_{1}$ and $\hat{u}^{(1)}_{1}$ lies in the coefficients of the basis function $\phi_{1}$. Specifically, the coefficient of $\phi_{1}$ for $\hat{u}^{(0)}_{1}$ is considerably smaller than that for $\hat{u}^{(1)}_{1}$. This suggests that the second solution, $\hat{u}^{(1)}_{1}$, may be formed by extending the first solution space ${\phi_{0}}$ through the inclusion of an adaptive basis function, $\phi_{1}$.
• Figure 4.3: Two solutions and its convergence for \ref{['eq4.1']}.
• Figure 4.4: Multiple solutions of \ref{['eq4.3']} by using our method.
• Figure 4.5: Multiple solutions of \ref{['eq4.5']} with different $(\lambda,\,p)$, and some spurious solutions (blue line) before applying filtering conditions.
• Figure 4.6: Basis functions computed by our algorithm for solving \ref{['eq4.5']} with $\lambda=1$ and $p=18$.