Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation
For researchers in numerical analysis and semiconductor simulation, this work provides rigorous convergence and stability guarantees for entropy-dissipative schemes, though the methods are extensions of known techniques.
The paper analyzes structure-preserving numerical schemes for a fourth-order quantum diffusion equation, proving entropy stability and second-order convergence for a BDF method, with numerical experiments showing exponential decay of entropies and Fisher information.
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.