Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation

TL;DR

This paper develops and analyzes structure-preserving numerical schemes for a complex quantum diffusion equation, ensuring entropy stability, nonnegativity, and convergence, with numerical evidence of exponential decay of key quantities.

Contribution

It introduces entropy-stable, entropy-dissipative numerical schemes for a fourth-order quantum diffusion equation, including a semi-discrete BDF method and a variational derivative approach, with proven stability and convergence.

Findings

The schemes preserve nonnegativity and dissipate entropy.

The semi-discrete solution converges with second-order accuracy in time.

Numerical experiments confirm exponential decay of entropy and Fisher information.

Abstract

Structure-preserving numerical schemes for a nonlinear parabolic fourth-order equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified entropy functional. The existence of a weak semi-discrete solution and, in a particular case, its temporal second-order convergence to the continuous solution is proved. The proofs employ an algebraic relation which implies the G-stability of the two-step BDF. Second, an implicit Euler and q-step BDF discrete variational derivative method are considered. This scheme, which exploits the variational structure of the equation, dissipates the discrete Fisher information (or energy). Numerical experiments show that the discrete (relative) entropies and Fisher information decay even exponentially fast to zero.