A Roadmap for Discretely Energy-Stable Schemes for Dissipative Systems Based on a Generalized Auxiliary Variable with Guaranteed Positivity

by   Zhiguo Yang, et al.

We present a framework for devising discretely energy-stable schemes for general dissipative systems based on a generalized auxiliary variable. The auxiliary variable, a scalar number, can be defined in terms of the energy functional by a general class of functions, not limited to the square root function adopted in previous approaches. The current method has another remarkable property: the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level, regardless of the time step sizes or the external forces. This property of guaranteed positivity is not available in previous approaches. A unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented. The discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems. The current method, termed gPAV (generalized Positive Auxiliary Variable), requires only the solution of linear algebraic equations within a time step. With appropriate choice of the operator in the algorithm, the resultant linear algebraic systems upon discretization involve only constant and time-independent coefficient matrices, which only need to be computed once and can be pre-computed. Several specific dissipative systems are studied in relative detail using the gPAV framework. Ample numerical experiments are presented to demonstrate the performance of the method, and the robustness of the scheme at large time step sizes.


gPAV-Based Unconditionally Energy-Stable Schemes for the Cahn-Hilliard Equation: Stability and Error Analysis

We present several first-order and second-order numerical schemes for th...

Second order, unconditionally stable, linear ensemble algorithms for the magnetohydrodynamics equations

We propose two unconditionally stable, linear ensemble algorithms with p...

Energy stable schemes for gradient flows based on novel auxiliary variable with energy bounded above

In this paper, we consider a novel auxiliary variable method to obtain e...

An Energy-Stable Scheme for Incompressible Navier-Stokes Equations with Periodically Updated Coefficient Matrix

We present an energy-stable scheme for simulating the incompressible Nav...

A gPAV-Based Unconditionally Energy-Stable Scheme for Incompressible Flows with Outflow/Open Boundaries

We present an unconditionally energy-stable scheme for approximating the...

Convergent and orthogonality preserving schemes for approximating the Kohn-Sham orbitals

To obtain convergent numerical approximations without using any orthogon...

Please sign up or login with your details

Forgot password? Click here to reset