Qiwei Feng

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 $Ω$.

Qiwei Feng

In this paper, we first consider linear 2D and 3D convection-diffusion-reaction equations $-\nabla\cdot (κ\nabla u) + {\bm v} \cdot \nabla u + λu = ϕ$ and $u_t - \nabla\cdot (κ\nabla u) + {\bm v} \cdot \nabla u + λu = ϕ$, where all $κ>0, {\bm v}, λ, ϕ$ are smooth variable functions. We derive fourth-order compact 9-point (2D) and 19-point (3D) finite difference methods (FDMs) to solve linear time-independent equations. As derivations of high-order compact FDMs are very complicated and involve cumbersome notation (especially in 3D), it is usually difficult for readers not specializing in high-order FDMs to follow derivations and replicate numerical results. In this paper, we observe interesting and novel expressions of stencils of high-order FDMs which introduce new restrictions of stencils to help construct compact fourth-order FDMs (2D and 3D) with easy, explicit, and short expressions. These simple stencils make the analysis of the truncation error easy for readers to understand and facilitate implementing proposed FDMs directly. For linear unsteady equations, we apply Crank-Nicolson (CN), BDF3, BDF4 methods with above compact FDMs to compute numerical solutions. Finally, we discuss nonlinear convection-diffusion-reaction equations in 2D and 3D, i.e., each function of $κ(u)>0, {\bm v}(u), λ(u)$ depends on the solution $u$. We linearize nonlinear equations by the fixed point method (iterative method) and use above simple fourth-order compact FDMs to solve linearized equations (unsteady equations also utilize CN, BDF3, and BDF4 methods). Each of proposed FDMs in 2D and 3D for linear, nonlinear, steady, and unsteady equations satisfies the discrete maximum principle and forms an M-matrix for the sufficiently small $h$, if the function $λ$ is nonnegative.