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WKB-based third order method for the highly oscillatory 1D stationary Schrödinger equation

Anton Arnold, Jannis Körner

TL;DR

The paper develops a third‑order, one‑step scheme for the highly oscillatory 1D Schrödinger equation $\varepsilon^{2}\varphi''+a(x)\varphi=0$ by applying a WKB‑based transformation that yields a smoother, coarse‑grid solvable system. Central to the method is solving a transformed $Z$‑system with Picard expansion up to $P=3$ and constructing accurate quadratures for the iterated oscillatory integrals via AM and SAM, producing explicit update matrices that achieve $\mathcal{O}_{\varepsilon,h}(\varepsilon^{3}h^{3}\max(\varepsilon,h))$ global error (under exact phase). The work also analyzes phase errors and presents a simplified third‑order variant that preserves the $\mathcal{O}(h^{3})$ convergence with reduced cost. Numerical results on Airy and other test problems demonstrate high efficiency and accuracy, including comparisons with Riccati defect correction, confirming the method’s practical value for small $\varepsilon$ where standard discretizations fail. The study advances uniformly accurate oscillatory solvers by extending prior second‑order WKB schemes to third order and providing rigorous error analysis and phase‑handling considerations.

Abstract

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Building upon the ideas from [Arnold, Ben Abdallah, Negulescu, SIAM J. Numer. Anal., 2011], we first analytically transform the given equation into a smoother (i.e. less oscillatory) equation. By developing sufficiently accurate quadratures for several (iterated) oscillatory integrals occurring in the Picard approximation of the solution, we obtain a one-step method that is third order w.r.t. the step size. The accuracy and efficiency of the method are illustrated through several numerical examples.

WKB-based third order method for the highly oscillatory 1D stationary Schrödinger equation

TL;DR

The paper develops a third‑order, one‑step scheme for the highly oscillatory 1D Schrödinger equation by applying a WKB‑based transformation that yields a smoother, coarse‑grid solvable system. Central to the method is solving a transformed ‑system with Picard expansion up to and constructing accurate quadratures for the iterated oscillatory integrals via AM and SAM, producing explicit update matrices that achieve global error (under exact phase). The work also analyzes phase errors and presents a simplified third‑order variant that preserves the convergence with reduced cost. Numerical results on Airy and other test problems demonstrate high efficiency and accuracy, including comparisons with Riccati defect correction, confirming the method’s practical value for small where standard discretizations fail. The study advances uniformly accurate oscillatory solvers by extending prior second‑order WKB schemes to third order and providing rigorous error analysis and phase‑handling considerations.

Abstract

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Building upon the ideas from [Arnold, Ben Abdallah, Negulescu, SIAM J. Numer. Anal., 2011], we first analytically transform the given equation into a smoother (i.e. less oscillatory) equation. By developing sufficiently accurate quadratures for several (iterated) oscillatory integrals occurring in the Picard approximation of the solution, we obtain a one-step method that is third order w.r.t. the step size. The accuracy and efficiency of the method are illustrated through several numerical examples.
Paper Structure (14 sections, 8 theorems, 92 equations, 10 figures)

This paper contains 14 sections, 8 theorems, 92 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Review of the WKB-based transformation from [2]
  3. Construction of the numerical method
  4. Approximation of the matrix M1
  5. Approximation of the matrix M2
  6. Approximation of the matrix M3
  7. Numerical scheme and error analysis
  8. Refined error estimate incorporating phase errors
  9. Simplified third order scheme
  10. Numerical results
  11. First example: Airy equation
  12. Comparison with results from program riccati
  13. Second example
  14. Conclusion

Key Result

Proposition 2.1

Arnold2011WKBBasedSF Let Hypothesis hypothesis_A be satisfied. Then the problem (Zsystem) has a unique solution $Z\in C^{6}(I)$ with the explicit form where the matrices $\mathbf{M}_{p}^{\varepsilon}$, $p\geq 1$ are given by Here, $\mathbf{I}$ denotes the $2\times 2$ identity matrix. Moreover we have the estimates with a constant $C>0$ independent of $\varepsilon$.

Figures (10)

  • Figure 1: Global errors $\max_{n\leq N}\lVert U(x_{n})-U_{n}\rVert_{\infty}$ for the Airy equation (\ref{['eqn:Airy']}) as functions of the step size $h$, for several $\varepsilon$-values. Left: WKB3 (solid lines with asterisks) and WKB3s (dashed lines with circles). Right: WKB2.
  • Figure 2: CPU times vs. global errors $\max_{n\leq N}\lVert U(x_{n})-U_{n}\rVert_{\infty}$ for the Airy equation (\ref{['eqn:Airy']}) on the spatial interval $[1, 2]$, for several $\varepsilon$-values. Left: WKB3 (solid lines with asterisks) and WKB3s (dashed lines with circles). Right: WKB2.
  • Figure 3: Global errors $\max_{n\leq N}\lVert U(x_{n})-U_{n}\rVert_{\infty}$ for the Airy equation (\ref{['eqn:Airy']}) on the spatial interval $[1, 2]$ as a function of the step size $h$, for several $\varepsilon$-values. Left: WKB3 (solid lines with asterisks) and WKB3s (dashed lines with circles). Right: WKB2.
  • Figure 4: CPU times vs. global errors $\max_{n\leq N}\lVert U(x_{n})-U_{n}\rVert_{\infty}$ for the Airy equation (\ref{['eqn:Airy']}) on the spatial interval $[1, 2]$ as a function of the step size $h$, for several $\varepsilon$-values. Left: WKB3 (solid lines with asterisks) and WKB3s (dashed lines with circles). Right: WKB2.
  • Figure 5: Global errors $\max_{n\leq N}\lVert U(x_{n})-\tilde{U}_{n}\rVert_{\infty}$ for the Airy equation (\ref{['eqn:Airy']}) on the spatial interval $[1, 2]$ as a function of the step size $h$, for several $\varepsilon$-values. The solid lines with asterisks correspond to WKB3, and the dashed lines with circles correspond to WKB3s. Here, we used an approximate phase $\tilde{\phi}$ computed with two different methods: Left: Composite Simpson's rule. Right: Clenshaw-Curtis algorithm based on $N_{cheb}=17$ Chebyshev grid points, with barycentric interpolation.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Proposition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Theorem 4.1
  • proof
  • ...and 4 more