Positivity-preserving and unconditionally energy stable numerical schemes for MEMS model

Abstract

In this paper, we propose and analyze the positivity-preserving, unconditionally energy stable, and linear second order fully discrete schemes for Micro-Electromechanical system (MEMS). More precisely, we use the first order backward difference formulation (BDF1) and second order Crank-Nicolson (CN) formulation for the temporal discretization, and the central finite difference method for spatial discretization. A variant of the exponential scalar auxiliary variable (ESAV) approach is involved in our numerical schemes to deal with the singular nonlinear term. The unconditional energy stability of the numerical schemes is rigorously proved, without any restriction for the time step sizes. Furthermore, we derive that the numerical solutions always preserve the positivity property of the MEMS model, that is the distance variable is always between 0 and the steady solution, at a point-wise level. A series of numerical simulations are presented to demonstrate the positivity and energy stability of our numerical schemes.