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An Integral Equation Method for Linear Two-Point Boundary Value Systems

Tianze Zhang, Yixuan Ma, Jun Wang

TL;DR

The paper develops an integral equation approach for linear two-point boundary value systems, with emphasis on choosing a background Green's function to yield a well-conditioned second-kind integral equation. It introduces two strategies to handle degenerate boundary conditions and demonstrates, via a conditioning case study, that the transformed formulations can provide robust stability. A Nyström discretization combined with a fast direct solver yields a black-box solver with linear scaling in the number of discretization points and high-order convergence. Numerical experiments across stiff and oscillatory problems show fast, accurate performance and solid robustness, indicating practical impact for large-scale boundary-value problems.

Abstract

We present an integral equation-based method for the numerical solution of two-point boundary value systems. Special care is devoted to the mathematical formulation, namely the choice of the background Green's function that leads to a well-conditioned integral equation. We then make use of a high-order Nystrom discretization and a fast direct solver on the continuous level to obtain a black-box solver that is fast and accurate. A numerical study of the conditioning of different integral formulations is carried out. Excellent performance in speed, accuracy, and robustness is demonstrated with several challenging numerical examples.

An Integral Equation Method for Linear Two-Point Boundary Value Systems

TL;DR

The paper develops an integral equation approach for linear two-point boundary value systems, with emphasis on choosing a background Green's function to yield a well-conditioned second-kind integral equation. It introduces two strategies to handle degenerate boundary conditions and demonstrates, via a conditioning case study, that the transformed formulations can provide robust stability. A Nyström discretization combined with a fast direct solver yields a black-box solver with linear scaling in the number of discretization points and high-order convergence. Numerical experiments across stiff and oscillatory problems show fast, accurate performance and solid robustness, indicating practical impact for large-scale boundary-value problems.

Abstract

We present an integral equation-based method for the numerical solution of two-point boundary value systems. Special care is devoted to the mathematical formulation, namely the choice of the background Green's function that leads to a well-conditioned integral equation. We then make use of a high-order Nystrom discretization and a fast direct solver on the continuous level to obtain a black-box solver that is fast and accurate. A numerical study of the conditioning of different integral formulations is carried out. Excellent performance in speed, accuracy, and robustness is demonstrated with several challenging numerical examples.

Paper Structure

This paper contains 12 sections, 13 theorems, 109 equations, 3 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction \newlabelsec:intro0
  2. Mathematical Preliminaries \newlabelsec:prelim0
  3. notation and definitions
  4. Formulation of integral equations
  5. Choices of background Green's functions
  6. A linear transform for degenerate boundary conditions
  7. Discretization and fast algorithms
  8. Recursive decomposition of the integral operator
  9. High-order Nyström discretization
  10. A case study of conditioning
  11. Numerical examples
  12. Conclusions

Key Result

Theorem 2.4

\newlabelthm:exist_unique0 Let $\Gamma(x)=(\gamma_1(x), \gamma_2(x),\cdots \gamma_n(x))$ be a fundamental matrix of the differential equation eqn:homo. Then the following properties are equivalent.

Figures (3)

  • Figure 1: The numerical result of Bessel equation.
  • Figure 1: Numerical solution of the viscous shock equation with $\epsilon=10^{-5}$. Dotted vertical lines represent endpoints of the subintervals
  • Figure 2: Example 2. CPU time consumed for each $p$ and error tolerance.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5: Green's function
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8: boundary condition matrix
  • Remark 2.9
  • Theorem 2.10: construction of Green's function via fundamental matrix
  • ...and 17 more