Optimal high-order diagonally-implicit Runge–Kutta schemes for nonlinear diffusive systems on atmospheric boundary layer

Abstract

Nonlinear diffusion equations are extensively applicable in diverse fields of science and engineering. Numerical stability is a common concern in this class of equations. In the present study, a three-stage third-order diagonally-implicit Runge–Kutta (DIRK) scheme is introduced by optimizing the error and linear stability analysis for a commonly used nonlinear diffusive system in atmospheric boundary layer. The proposed scheme is stable for a wide range of time steps and able to resolve different diffusive systems with diagnostic turbulence closures, or prognostic ones with a diagnostic length scale, with enhanced accuracy and stability compared to current schemes. It maintains A-stability, which makes it appropriate for the solution of stiff problems. The procedure implemented in this study is quite general and can be used in other diffusive systems as well.