Preprint: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport

20 December 2023

Together with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré we propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size.

Logarithmic plot of numerical solution the DLSS equation started from a discretization of Z_m,ε^-1[ε^1/2+ (1+cos(2π x)/2)^m]^2 with parameters ε=0.001, m=8,16. — Logarithmic plot of numerical solution the DLSS equation started from a discretization of $Z_{m,\varepsilon}^{-1} \bigl[\varepsilon^{1/2}+ \bigl(\frac{1+\cos(2\pi x)}{2}\bigr)^{\!m}\bigr]^{\!2}$ with parameters $\varepsilon=0.001$ , $m=8,16$ .

Now published in Numer. Math. 157(4), 1347–1395 (2025).

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