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.
An energy-dissipative level-set method for the incompressible two-phase Navier-Stokes equations with surface tension using functional entropy variables
A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
Institute of Mathematics, University of Zurich（苏黎世大学数学研究所）
Entropy dissipative high order schemes for a hyperbolic model of two-layer thin film flow
University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Chair of Applied Mathematics（斯图加特大学应用分析与数值模拟研究所应用数学教席）；Clausthal University of Technology, Institute of Mathematics, Numerical Analysis Research Group（克莱施塔尔工业大学数学研究所数值分析研究组）；Indian Institute of Technology Delhi, Department of Mathematics（德里印度理工学院数学系）
Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization
