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

Abstract

Abstract. Structure-preserving numerical schemes for a nonlinear parabolic fourthorder 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.

Long-time behavior of a finite volume discretization for a fourth order diffusion equation

“…Several (semi-)discrete approximations of (1) have been studied, both analytically and numerically. The schemes presented in [4,7,15,28,34] inherit some structural properties of (1), like monotonicity of certain quantities. All of these schemes have in common that they provide nonnegative (semi-)discrete solutions.…”

Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

“…In fact, the above theorem is a generalization of Theorem 2 in [32], which is proved for α = 1 and the two-step BDF method only. Another example is α = 4/3 and the fast-diffusion operator A(u) = − (u 1/3 ) with Dirichlet boundary conditions, although we do not study this operator here.…”

“…Numerical examples for the quantum diffusion equation can be found in [32]. We choose the two-step BDF and γ -method in time, defined in Remark 4, and finite differences in space.…”

A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation

“…where H 0 was defined in (6), and z = z δ [ x]. Naturally, H δ inherits non-negativity, and vanishes only for x with x k = a + (b − a)k/K.…”

“…At the very least, the scheme should produce non-negative (preferably positive) discrete solutions, but there is no reason to expect that behavior from a standard discretization approach. Several (semi-)discretizations for (1)-(3) that guarantee positivity have been proposed in the literature [6,10,34,36]. In all of them, positivity actually appears as a consequence of another, more fundamental feature: each of these schemes also inherits a Lyapunov functional, either a logarithmic/power-type entropy [6,10,34], or a variant of the Fisher information [6,20,36].…”