$L^p$ Estimates for Numerical Approximation of Convex Hamilton-Jacobi Equations

Technical Summary: $L^p$ Error Estimates for Numerical Approximation of Convex Hamilton–Jacobi Equations

Problem Statement
The paper addresses the numerical approximation of convex Hamilton–Jacobi (HJ) equations on the $d$-dimensional flat torus $\mathbb{T}^d$. The specific equation under consideration is:
$$\begin{cases} \partial_t u + H(Du) = 0 & \text{in } Q := \mathbb{T}^d \times (0, T), \\ u(x, 0) = g(x) & \text{on } \mathbb{T}^d, \end{cases}$$
where the Hamiltonian $H$ is smooth, convex, and coercive, and the initial datum $g$ is Lipschitz continuous and semiconcave. The authors focus on two classes of monotone numerical schemes: finite-difference methods and semi-Lagrangian methods. While classical results (e.g., Crandall–Lions, Barles–Souganidis) establish $L^\infty$ convergence rates of order $1/2$ for general monotone schemes without convexity assumptions, this work seeks to establish sharper error estimates in $L^p$ norms ($1 \le p < +\infty$) by exploiting the convexity of the Hamiltonian and the semiconcavity of the solution.

Methodology
The core methodology relies on the adjoint method combined with semiconcavity estimates. The approach diverges from standard viscosity solution techniques (which typically rely on the doubling-of-variables technique) by utilizing a duality argument originally developed for scalar conservation laws and adapted for HJ equations.

1. Adjoint Framework: The authors couple the nonlinear HJ equation with a linearized adjoint equation. For a finite-difference scheme, the error $z = u^\varepsilon - u$ satisfies a transport-type equation with a source term representing the truncation error. By multiplying this equation by the solution $\rho$ of the backward adjoint equation (with terminal data $\text{sgn}(z)$), the $L^1$ norm of the error is expressed as an integral involving the truncation error and the adjoint solution.
2. Semiconcavity Preservation: A critical component of the analysis is proving that the numerical schemes preserve the semiconcavity of the initial datum.
   • For finite-difference schemes, the authors construct a continuous-space, discrete-time auxiliary scheme. They demonstrate that the monotone update operator preserves semiconcavity in the continuous sense, avoiding the need for discrete semiconcavity reconstruction.
   • For semi-Lagrangian schemes, the analysis utilizes the dynamic programming representation of the discrete value functions. The proof leverages the Lipschitz continuity and semiconcavity properties of these value functions, along with the differentiability of the solution almost everywhere (linked to the uniqueness of the optimal control).
3. Error Decomposition: The truncation error is decomposed into two parts:
   • The error due to time interpolation (between discrete time steps).
   • The spatial consistency error (difference between the discrete gradient and the continuous gradient).
     Using the preserved semiconcavity, the authors bound the $L^1$ norm of the second derivatives of the numerical solution, which allows them to control the spatial consistency error.
4. Interpolation: Once an $L^1$ error bound of order 1 is established, the authors employ interpolation inequalities between the $L^1$ and the classical $L^\infty$ estimates (order $1/2$) to derive $L^p$ error bounds for all $1 \le p < +\infty$.

Key Contributions and Results

• Finite-Difference Schemes:

  • The authors prove an $L^1$ error estimate of order $O(h + \Delta t)$ for monotone finite-difference schemes using a consistent, convex numerical Hamiltonian.
  • By interpolating with the standard $L^\infty$ estimate of order $O((h+\Delta t)^{1/2})$, they derive the $L^p$ error bound:
    $$\|u^\varepsilon_n - u(\cdot, n\Delta t)\|_{L^p} \le C (h + \Delta t)^{\frac{1}{2} + \frac{1}{2p}}.$$
  • This result extends previous work (e.g., Cagnetti–Gomes–Tran) to multidimensional settings and a broader class of convex monotone Hamiltonians, without relying on a transport–projection structure.

• Semi-Lagrangian Schemes:

  • For semi-Lagrangian schemes with Hamiltonians of the form $H(x, p) = H_0(p) - V(x)$, the authors establish an $L^1$ error estimate of order $O(\Delta t)$ for the time-discrete scheme.
  • For the fully discrete scheme (involving spatial grid interpolation), the $L^1$ error bound is $O(\Delta t + h/\Delta t)$.
  • The corresponding $L^p$ convergence rate is derived as:
    $$\| \tilde{u}^\varepsilon_n - u(\cdot, n\Delta t) \|_{L^p} \le C \left( \Delta t + \frac{h}{\Delta t} \right)^{\frac{1}{p}} \left( \Delta t^{\frac{1}{2}} + \frac{h}{\Delta t} \right)^{1-\frac{1}{p}}.$$

Significance and Claims
The paper claims to provide a systematic derivation of $L^p$ error estimates for convex HJ equations, improving upon the standard $L^\infty$ rates known for general monotone schemes. The significance lies in:

1. Sharpening Convergence Rates: Demonstrating that under convexity and semiconcavity assumptions, the convergence rate in $L^1$ (and consequently in $L^p$ for $p>1$) is of order 1, which is superior to the $1/2$ order typically found in $L^\infty$.
2. Methodological Adaptation: Successfully adapting the adjoint method, traditionally used for conservation laws and vanishing viscosity approximations, to analyze local monotone schemes (Crandall–Lions type) and semi-Lagrangian schemes directly.
3. Generalization: Extending the applicability of these estimates to multidimensional tori and general classes of convex numerical Hamiltonians, rather than being restricted to specific one-dimensional or transport-projection structures.

The authors maintain a modest tone, noting that their results rely on standard assumptions (smoothness, convexity, coercivity of $H$; Lipschitz and semiconcavity of $g$) and that the $L^1$ mechanism is a direct application of the semiconcavity-duality principle, with the novelty residing in the verification of these hypotheses for the specific numerical schemes considered.