Derivation of resonance-based schemes via normal forms

Technical Summary: Derivation of Resonance-Based Schemes via Normal Forms

Problem Statement
The paper addresses the numerical approximation of dispersive partial differential equations (PDEs), specifically focusing on equations with low regularity initial data, such as the cubic nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries (KdV) equation. Classical numerical integrators and splitting schemes typically require high regularity of the solution to achieve optimal convergence rates. While "resonance-based schemes" have been proposed to mitigate this requirement by embedding resonance structures into the discretization, a rigorous algebraic connection between these schemes and the classical "normal forms" approach (specifically Poincaré-Dulac and Birkhoff normal forms) used in the analytical study of dispersive equations remained an open question. The primary goal is to systematically derive low regularity schemes by leveraging the Poincaré-Dulac normal form, thereby unifying the analytical normal form theory with numerical discretization.

Methodology
The authors employ a systematic derivation combining combinatorial algebra, Hopf algebras, and Fourier analysis. The methodology proceeds through the following key steps:

1. Fourier Decorated Trees: The iterated integrals arising from the Duhamel formulation of the PDE are encoded using a specific class of decorated trees. These trees carry frequency decorations (nodes) and operator decorations (edges) corresponding to the polynomial nonlinearity and the linear dispersive operator.
2. Algebraic Tools (Hopf Algebras): The derivation utilizes a variant of the Butcher-Connes-Kreimer (BCK) coproduct and a grafting product. Crucially, the paper introduces an arborification map ($a$) derived from mould calculus. This map transforms expansions in terms of rooted trees (words) into a form suitable for normal form analysis.
3. Poincaré-Dulac Normal Form: The core analytical step involves the Poincaré-Dulac normal form, which eliminates non-resonant monomials by integrating by parts. The phase function $\Phi(\bar{k})$ associated with the oscillatory integrals is decomposed into a dominant part ($\Phi_{dom}$) and a lower part ($\Phi_{low}$).
4. Low Regularity Discretization: Unlike standard normal forms that might require full Taylor expansions (demanding high regularity), this approach performs a Taylor expansion only on the lower part ($\Phi_{low}$) while treating the dominant part ($\Phi_{dom}$) exactly via integration by parts. This splitting allows the scheme to handle solutions with lower regularity. The dominant part is chosen such that its inverse can be mapped back to physical space using standard differential operators.
5. Forest Formula and Inductive Construction: The paper derives a forest formula for the low regularity schemes. This is achieved by iterating the BCK coproduct and the arborification map. The scheme is constructed inductively, discretizing the oscillatory integrals associated with subtrees containing the root and iterating the process on the remaining disconnected parts.

Key Contributions

• Systematic Derivation: The paper provides the first systematic derivation of resonance-based low regularity schemes directly from the Poincaré-Dulac normal form, bridging the gap between analytical normal form theory and numerical discretization.
• Explicit Formulae: The authors derive explicit formulae for the coefficients of the numerical schemes and, crucially, for their local error. The scheme is expressed as a sum over decorated trees with coefficients determined by the arborification map and the specific splitting of the phase.
• Forest Formula: A new forest formula is introduced to describe the low regularity schemes. This formula arises from an iteration of a Butcher-Connes-Kreimer type coproduct, an algebraic ingredient previously absent in normal form derivations.
• Arborification and Mould Calculus: The work integrates the arborification map from mould calculus into the numerical scheme derivation, allowing for the rewriting of expansions in terms of words and facilitating the handling of combinatorial factors arising from repeated integration by parts and Taylor expansions.

Results

• Theorem 5.10: This is the central result, stating that the proposed scheme $(\Pi^{n,r}T)(t)$ approximates the exact oscillatory integral $(\Pi T)(t)$ with a local error of order $O(t^{r+1} E_{n,r}(T))$. The error term $E_{n,r}(T)$ is explicitly defined in terms of the lower parts of the Fourier operators.
• Corollary 5.11: This corollary translates the tree-level result into a full numerical scheme $U^{n,r}_k(t, u_0)$ for the PDE solution. It establishes that the local error of the scheme is a sum of terms involving the elementary differentials and the explicit error coefficients derived in Theorem 5.10.
• Equivalence under Assumption 1: Under a mild assumption (Assumption 1) regarding the consistency of dominant part computations across different schemes, the paper demonstrates that the local error of these normal-form-derived schemes is comparable to that of the low regularity schemes previously proposed in [13].
• Example Verification: The paper verifies the derivation on the cubic NLS equation, showing that for specific cases (e.g., $n=2, r=2$), the derived scheme and its local error match those obtained in previous works ([13], [36]), confirming the validity of the approach.

Significance and Claims
The paper claims that this work resolves an open question regarding the connection between resonance-based schemes and normal forms. By using the Poincaré-Dulac normal form as a foundation, the authors provide a rigorous algebraic framework for generating low regularity schemes.

The significance lies in:

1. Unification: It unifies the analytical study of dispersive equations (via normal forms) with numerical analysis, showing that the "trick" of splitting the phase and integrating by parts is a natural consequence of the normal form procedure.
2. Explicitness: It offers explicit formulae for the schemes and their errors, moving beyond abstract existence results to concrete, implementable algorithms.
3. Generality: While the derivation is illustrated on NLS, the framework is presented for general dispersive equations with polynomial nonlinearities.

The authors remain modest regarding the scope of their claims. They note that while the schemes share similar local error properties with existing methods under Assumption 1, it is not immediately clear if the schemes coincide exactly in the general case without specific assumptions. Furthermore, they acknowledge that extending these results to symplectic schemes via the Birkhoff normal form remains an open and challenging problem due to the rigidity of symplectic transforms, whereas the Poincaré-Dulac approach (relying on integration by parts) is more flexible for this purpose. The paper does not claim to solve global stability or long-time behavior issues, noting that such analysis requires discrete Strichartz estimates beyond its current scope.