Convergence of a scheme for a two dimensional nonlocal system of transport equations

In this paper, we numerically study a two-dimensional system modeling the dynamics of dislocation densities. This system is hyperbolic, but not strictly hyperbolic, and couples two non-local transport equations. It is characterized by weak regularity in both the velocity and the initial data. We propose a semi-explicit finite difference (IMEX) numerical scheme for the discretization of this system, after regularizing the singular velocity using a Fejér kernel. We show that this scheme preserves, at the discrete level, an entropy estimate on the gradient, which then allows us to establish the convergence of the discrete solution to the continuous solution. To our knowledge, this is the first convergence result obtained for this type of system. We conclude with some numerical illustrations highlighting the performance of the proposed scheme.

💡 Research Summary

The paper addresses the numerical analysis of a two‑dimensional nonlocal system that models the evolution of dislocation densities in crystalline materials. The system consists of two coupled transport equations for the scalar fields ρ⁺ and ρ⁻, each representing the plastic distortion associated with dislocations of opposite Burgers vectors. The velocity field is given by a nonlocal term involving the composition of two periodic Riesz transforms, which can be expressed as a convolution with a singular kernel K. Because K is highly singular and the initial data have only low regularity (∂₁ρ⁺, ∂₁ρ⁻ belong merely to the Zygmund space L log L), the standard theory for hyperbolic systems does not apply, and even the existence of weak solutions requires a delicate entropy estimate.

The authors first regularize the singular kernel by convolving it with a two‑dimensional Fejér kernel F₂M, obtaining a smoothed kernel σ_KM = F₂M * K. This regularization is equivalent to taking a Cesàro mean of order M of the Fourier series of K, and it preserves the positivity of the Fourier coefficients, which is crucial for the entropy analysis. The initial data are regularized in the same way, yielding smooth periodic functions ρ⁺₀ᴹ, ρ⁻₀ᴹ.

A semi‑explicit (IMEX) finite‑difference scheme is then introduced. Time discretization uses an explicit Euler step for the transport part and an implicit treatment of the nonlocal term, while space discretization employs an upwind finite‑difference operator for the derivative ∂₁. The scheme can be written schematically as
ρ_i^{n+1} = ρ_i^{n} – Δt a(tⁿ)(∂₁ρ_i^{n}+L) – Δt