Splitting schemes for unsteady problems involving the grad-div operator

TL;DR

This paper explores splitting schemes for unsteady problems with the grad-div operator, aiming to develop unconditionally stable, decoupled methods suitable for Cartesian grids, supported by theoretical stability analysis and numerical examples.

Contribution

It introduces unconditionally stable splitting schemes that decouple components in unsteady grad-div problems, extending the stability theory of additive operator-difference schemes.

Findings

Schemes are unconditionally stable.

Decoupling of equations is achieved on Cartesian grids.

Numerical example confirms theoretical stability.

Abstract

In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the solution vector. In this paper we discuss various possibilities to decouple the equations for the different components that result in unconditionally stable schemes. If the spatial discretization uses Cartesian grids, the resulting schemes are Locally One Dimensional (LOD). The stability analysis of these schemes is based on the general stability theory of additive operator-difference schemes developed by Samarskii and his collaborators. The results of the theoretical analysis are illustrated on a 2D numerical example with a smooth manufactured solution.