Fourth-order compact finite difference methods for 2D and 3D nonlinear convection-diffusion-reaction equations

Abstract: 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.