Error analysis of the Lie splitting for semilinear wave equations with finite-energy solutions

1. Problem Statement

The paper addresses the numerical time integration of the semilinear wave equation on the full space $\mathbb{R}^3$:
$$\partial_t^2 u = \Delta u - \mu |u|^{\alpha-1}u, \quad (t, x) \in [0, T] \times \mathbb{R}^3$$
with finite-energy initial data $(u_0, v_0) \in \dot{H}^1(\mathbb{R}^3) \times L^2(\mathbb{R}^3)$.

• Regularity: The focus is on solutions with finite energy (regularity $\dot{H}^1 \times L^2$). This is a low-regularity regime where standard Sobolev embeddings do not guarantee $L^\infty$ bounds on the solution (which requires $H^s$ with $s > 3/2$).
• Nonlinearity Exponent: The analysis covers the range $\alpha \in [3, 5]$.
  • Subcritical: $\alpha < 5$.
  • Energy-Critical: $\alpha = 5$. This case is significantly more challenging due to the scaling invariance of the equation and the lack of uniform spacetime bounds.
• Goal: To establish rigorous error bounds for time-discretization schemes (specifically Lie splitting) in the $L^2 \times \dot{H}^{-1}$ norm, a setting previously unexplored for scaling-critical dispersive problems.

2. Methodology

The authors employ a sophisticated analysis based on discrete-time Strichartz estimates combined with frequency cut-offs.

A. Frequency Filtering

Standard time-stepping schemes fail to converge optimally in this low-regularity setting without modification. The authors introduce a frequency cut-off operator $\pi_K = \mathcal{F}^{-1} \mathbf{1}_{\{|\xi| \le K\}} \mathcal{F}$ into the numerical scheme.

• The cut-off level $K$ is chosen as a function of the time step $\tau$ (e.g., $K = \tau^{-1}$ or $K = \tau^{-3/2}$).
• This filtering regularizes the nonlinearity, allowing the use of discrete Strichartz estimates which would otherwise fail at "forbidden endpoints" (like $L^2_t L^\infty_x$).

B. Discrete Strichartz Estimates

A core theoretical contribution is the derivation of discrete-time Strichartz estimates for the wave equation.

• Standard Estimates: The authors prove that for wave-admissible triples $(p, q, \gamma)$, the discrete solution satisfies estimates of the form:
  $$\|\pi_K e^{in\tau|\nabla|} f\|_{\ell^p_\tau L^q} \lesssim (K\tau)^{1/p} \|f\|_{\dot{H}^\gamma}$$
• Forbidden Endpoint: Crucially, they derive estimates for the endpoint $(p, q) = (2, \infty)$, which is known to fail in continuous time without modification. Their discrete version includes a logarithmic correction:
  $$\|\pi_K e^{in\tau|\nabla|} f\|_{\ell^2_\tau L^\infty} \lesssim \sqrt{K\tau + \log(1 + KN\tau)} \|f\|_{\dot{H}^1}$$
  This result is essential for handling the critical case and higher-order schemes.

C. Error Analysis Strategy

The error analysis follows a "double induction" strategy:

1. Local Error Decomposition: The global error is decomposed into local truncation errors and projection errors.
2. Induction on Time Intervals: The time interval $[0, T]$ is split into smaller sub-intervals where the Strichartz norms of the solution are small.
3. Recursive Bounds: The error is bounded recursively on these sub-intervals, utilizing the discrete Strichartz estimates to control the nonlinear terms without requiring $L^\infty$ bounds on the solution.

3. Key Contributions and Results

A. First-Order Convergence for Lie Splitting

For the standard Lie splitting scheme (with frequency filter $\pi_{\tau^{-1}}$), the authors prove first-order convergence in the $L^2 \times \dot{H}^{-1}$ norm for both subcritical and critical cases.

• Theorem 1.1 (Subcritical $\alpha < 5$):
  $$\|U(n\tau) - U_n\|_{L^2 \times \dot{H}^{-1}} \le C\tau$$
  The constant $C$ and step size $\tau_0$ depend on the solution norm and $T$.
• Theorem 1.2 (Critical $\alpha = 5$):
  $$\|U(n\tau) - U_n\|_{L^2 \times \dot{H}^{-1}} \le C\tau$$
  Significance: This is the first error analysis for a time discretization of a scaling-critical dispersive problem with finite-energy data. The step size $\tau_0$ depends on the specific solution (not just its norm), reflecting the delicate nature of the critical case.

B. Corrected Lie Splitting with Order 3/2

For the cubic case ($\alpha = 3$), the authors analyze a corrected Lie splitting scheme. This scheme adds a correction term involving the operator $\phi_2(-2\tau A)$ to cancel second-order derivative terms in the local error.

• Filter Level: They choose a higher cut-off $K = \tau^{-3/2}$ to balance the truncation error and the Strichartz loss.
• Theorem 1.3: They prove a convergence order of $3/2$:
  $$\|U(n\tau) - U_n\|_{L^2 \times \dot{H}^{-1}} \le C\tau^{3/2}$$
• Optimality: The loss of the factor $\tau^{1/2}$ from the optimal second order is attributed to the logarithmic loss in the endpoint Strichartz estimates. The authors argue this result is likely optimal for this regularity class.

4. Significance and Impact

1. Novelty in Critical Regimes: Prior to this work, error analysis for semilinear wave equations was largely restricted to higher regularity ($H^2$) or subcritical cases where $L^\infty$ bounds were available. This paper breaks new ground by handling the energy-critical case ($\alpha=5$) with minimal regularity ($\dot{H}^1$).
2. Discrete Strichartz Estimates: The derivation of discrete Strichartz estimates with logarithmic corrections for the forbidden endpoint $(2, \infty)$ is a major technical achievement. These tools are likely to be applicable to other dispersive PDEs (like Schrödinger equations) in low-regularity settings.
3. Frequency Filtering as a Tool: The paper demonstrates that frequency filtering is not just a regularization technique but a necessary component for achieving convergence in scaling-critical problems where standard schemes fail due to the lack of uniform bounds.
4. Optimality of Corrected Schemes: The analysis of the corrected Lie splitting provides a blueprint for constructing higher-order schemes in low-regularity regimes, showing that while formal order 2 is impossible due to Strichartz losses, order $3/2$ is achievable and optimal.

5. Conclusion

Ruff and Schnaubelt provide a rigorous mathematical foundation for the numerical simulation of semilinear wave equations with finite energy. By combining frequency filtering with novel discrete Strichartz estimates, they establish convergence rates for Lie splitting schemes in regimes previously considered analytically intractable for numerical analysis. Their work bridges the gap between the well-posedness theory of critical dispersive equations and their numerical approximation.