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Low regularity exponential-type integrators for semilinear Schr\"odinger equations
Alexander Ostermann, Katharina Schratz
TL;DR
The paper addresses first-order exponential-type integration for quadratic Schrödinger equations without added solution regularity. It derives schemes for the one-dimensional equations and proves first-order convergence without additional smoothness; the practical Fourier-space construction has a dimensional constraint.
Problem
The paper addresses how to obtain first-order exponential-type integrators for one-dimensional quadratic Schrödinger equations without additional solution regularity.
Source: §3 · pp. 11, 14Method
It derives exponential-type integrators for the one-dimensional quadratic equations (27) and (28).
Source: §3 · p. 11Results
For equation (27), the integrator is first-order convergent in H^r without additional smoothness assumptions on the exact solution.
Source: §3 · p. 14Takeaways & Limitations
The section establishes that these one-dimensional quadratic equations admit first-order exponential-type integrators without additional regularity assumptions.
Source: §3 · p. 11The described convolution-based practical implementation is limited to dimension one because it must be carried out fully in Fourier space.
Source: §1.1 · p. 5
Abstract
from arXiv · showhide
We introduce low regularity exponential-type integrators for nonlinear Schr\"odinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order convergence in $H^r$ for solutions in $H^{r+1}$ ($r>d/2$) of the derived schemes. This allows us lower regularity assumptions on the data in the energy space than for instance required for classical splitting or exponential integration schemes. For one dimensional quadratic Schr\"odinger equations we can even prove first-order convergence without any loss of regularity. Numerical experiments underline the favorable error behavior of the newly introduced exponential-type integrators for low regularity solutions compared to classical splitting and exponential integration schemes.
1. Low regularity exponential integrators
The section develops exponential-type integrators for nonlinear Schrödinger equations using a twisted variable and Fourier-based approximations. It extends the construction beyond the one-dimensional cubic case while identifying a practical implementation constraint.
1.1.1. Cubic nonlinearities p = 1 in dimension d = 1
- The convolution in the one-dimensional cubic construction cannot be straightforwardly converted to physical space, so practical iteration must remain in Fourier space.The passage identifies this as the reason the practical scheme is limited to dimension one.Source: §1.1 · p. 5
- The one-dimensional cubic remainder is bounded by cτ^(1+γ)∥w∥_(r+γ)^3 for 0≤γ≤1 and r>1/2.This bound supports the resulting scheme’s low-regularity error analysis.Source: §1.1 · p. 6
1.1. Cubic nonlinearities p = 1
- The cubic construction is based on iterating Duhamel’s formula in the twisted variable, and yields an exponential-type integration scheme.The section also contrasts its approximation with the classical exponential integrator, whose approximation error requires additional smoothness.Source: §1 · p. 4, §1.1 · p. 7
1.1.2. Cubic nonlinearities p = 1 in dimension d ≥1
- The exponential-type integrator (20) is generalized to dimensions d ≥ 1, with the stated remainder bound holding for 0 ≤ γ ≤ 1 and r > d/2.The dominant quadratic term 2j2 is treated exactly.Source: §1.1 · p. 7
1.2. Nonlinearities with integer p
- For general integer p, the section derives a numerical scheme and states its exponential-type form in the original variable.The construction first expresses the integral using Fourier expansion.Source: §1.2 · p. 8
2. Error analysis
The error analysis combines local-error and stability estimates to obtain global error bounds for the exponential-type scheme. Its convergence results require the exact solution to retain the stated Sobolev regularity over the time interval.
Local error
- The local-error estimate assumes the exact flow remains in H^(r+γ) during a time step, for r>d/2 and 0≤γ≤1.The estimate’s constant depends on the flow’s H^(r+γ) norm over that step.Source: §2 · p. 9
Global convergence
- The proof combines the local-error bound with stability and iterates the resulting estimate to establish global convergence.The argument uses induction or a Lady Windermere’s fan argument.Source: §2 · pp. 9–10
- Theorem 2.3 gives a global-error bound for r>d/2 when the exact solution remains in H^(r+γ), with 0<γ≤1, through the time interval.The bound’s constant depends on the supremum of the solution’s H^(r+γ) norm.Source: §2 · p. 9
- The corresponding result for the original Schrödinger solution assumes u(t)∈H^(r+γ) and gives a global-error bound for the scheme.The corollary follows from the twisted-variable theorem.Source: §2 · p. 10
3. Quadratic Schr¨odinger equations in one space dimension
For one-dimensional quadratic Schrödinger equations, the section derives exponential-type integrators and analyzes their errors. It establishes first-order convergence without additional smoothness assumptions on the exact solution.
3.1. Quadratic Schrödinger equations of type (27)
- The section derives first-order exponential-type integrators for Schrödinger equations of types (27) and (28), without imposing any additional regularity assumptions on the solution.Source: §3 · p. 11
3.3. Error analysis
- For equation (27), the local-error lemma assumes only that the exact flow remains in H^r, where r>1/2.The lemma’s constant depends on the flow’s H^r norm during the step.Source: §3 · p. 13
- The global convergence theorem for scheme (34) assumes the exact solution remains in H^r, for r>1/2.The proof relies on the local-error and stability estimates.Source: §3 · p. 14
- The integrator for equation (27) is first-order convergent in H^r without additional smoothness assumptions on the exact solution, for r>1/2.The stated result assumes the solution remains in H^r over the time interval.Source: §3 · p. 14
- The section also states first-order convergence results for the integrator approximating equation (28), under an H^r solution assumption for r>1/2.The result is stated in Corollary 3.5.Source: §3 · p. 14
4. Numerical experiments
The numerical experiments compare the new exponential-type integrators with classical splitting and exponential integration schemes. Across several equation types and regularity settings, the new schemes retain favorable error behavior where classical methods may lose order or fail to converge.
4. Numerical experiments
- The experiments report that the new integrator is preferable for low-regularity solutions and remains first-order convergent for rough solutions, while classical methods can suffer order reduction.The study uses a Fourier pseudospectral space discretization with largest Fourier mode K = 210.Source: §4 · p. 15
4.1.1. Cubic Schr¨odinger equations. —
- For cubic equations, the new scheme is first-order convergent for all ϑ ≥ 2; Lie splitting, Strang splitting, and classical exponential integration lose regularity or order at smaller ϑ.The classical schemes show their expected orders at higher ϑ, while the new scheme also has small errors at ϑ = 3.Source: §4.1.1 · p. 16, Figure 2 · p. 17, Figure 3 · p. 17
4.1.2. Quintic Schr¨odinger equations
- For non-smooth quintic data, the new scheme is first-order convergent, while classical exponential integration does not converge and both splitting methods fall to order one half.The cited comparison measures discrete L2 error at T = 1.Source: §4.2 · p. 20, Figure 7 · p. 21
- With small nonlinearity, only the new scheme is reported as both very accurate and first-order convergent for non-smooth initial data.The experiment uses µ = 0.01 and T = 10; for smooth data, all methods retain their classical orders, but the new scheme is most accurate over the tested step sizes.Source: §4.2 · p. 20, Figure 8 · p. 21
4.3. Numerical experiments for semilinear Schr¨odinger equations with non-integer
- For non-integer p, splitting convergence deteriorates below p = 1, whereas both exponential integrators retain first-order convergence down to very small p.The authors identify this behavior as a subject for future research.Source: §4.3 · pp. 21–22