Peter Minev

Jean-Luc Guermond, Peter Minev

In this paper we present extensions of the schemes proposed in \cite{GM14} that lead to a decoupling of the velocity components in the momentum equation. The new schemes reduce the solution of the incompressible Navier-Stokes equations to a set of classical uncoupled parabolic problems for each Cartesian component of the velocity. The pressure is explicitly recovered after the velocity is computed.

Peter Minev, Petr N. Vabishchevich

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.

Qiwei Feng, Bin Han, Peter Minev

In this paper, we discuss the 2D convection-diffusion-reaction equation with variable smooth coefficients and the Dirichlet boundary condition on a complicated, thin, and curved domain. We propose the fourth-order compact FDM at every grid point with the uniform Cartesian mesh. For the regular stencil center, we utilize the fourth-order compact 9-point FDM to approximate the solution. According to the preliminary analysis, we use vertical and horizontal transformations to derive fourth-order compact FDMs in 10 cases for all irregular stencil centers. To obtain the left-hand side of the stencil of the fourth-order FDM in each case, we only need to solve an at most $6 \times 24$ linear system which is presented with the explicit formula. The right-hand side of the FDM is constructed in explicit expression for any irregular stencil centers too. To achieve the fourth-order consistency, up to second-order partial derivatives of convection, diffusion, reaction, and source terms are used for the FDM at the regular stencil center, and the FDM at an irregular stencil center only requires first-order partial derivatives of convection, diffusion, reaction, and source terms, and up to third-order derivatives of the Dirichlet boundary function and the parametric expression of the boundary curve. We test challenging domains with 100-leaf, high-curvature, high-frequency, sharply varying, and nearly overlapping boundary curves, the proposed FDM produces the high accuracy and the stable fourth-order convergence rate in $l_2$ and $l_{\infty}$ norms. All stencils of our FDMs have a simple desired structure by only keeping grid points inside $Ω$ in the standard compact 9-point stencil for both regular stencils and boundary stencils, but without assuming any information outside the domain $Ω$.