Strongly Stable High-order Temporal Integration Schemes for Nonlinear Systems in Atmospheric Models

Nonlinearity accompanied with stiffness in atmospheric boundary layer physical parameterizations is a well-known concern in numerical weather prediction (NWP) models. Nonlinear diffusion equations, furthermore, are a class of equations which are extensively applicable in different fields of science and engineering. Numerical stability and accuracy is a common concern in this class of equation.
In the present research, a comprehensive effort has been made toward the temporal integration of such equations. The main goal is to find highly stable and accurate numerical methods which can be used specifically in atmospheric boundarylayer simulations in weather and climate prediction models, and extensively in other models where nonlinear differential equations play an important role, such as Navier-Stokes equations.
A modified extended backward differentiation formula (ME BDF) scheme is adapted and proposed at the first stage of this research. Various aspects of this scheme, including stability properties, linear stability analysis, and numerical experiments, are studied with regard to applications for the time integration of commonly used nonlinear damping and diffusive systems in atmospheric boundary layer models. A new temporal filter which leads to significant improvement of numerical results is proposed. Nonlinear damping and diffusion in the turbulent mixing of the atmospheric boundary layer is dealt with in the next stage by using optimally stable singly diagonally-implicit Runge-Kutta (SDIRK) methods, which have been proved to be effective and computationally efficient for the challenges mentioned in the literature.
Numerical analyses are performed, and two schemes are modified to enhance their numerical features and stability.
Three-stage third-order diagonally-implicit Runge-Kutta (DIRK) scheme is introduced by optimizing the error and linear stability analysis for the aforementioned nonlinear diffusive system. The new scheme is stable for a wide range of time steps and is 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. The procedure implemented in this study is quite general and can be used in other diffusive systems as well. As an extension of this study, high-order low-dissipation low-dispersion diagonally implicit Runge-Kutta schemes are analyzed and introduced, based on the
optimization of amplification and phase errors for wave propagation, and various optimized schemes can be obtained. The new scheme shows no dissipation. It is illustrated mathematically and numerically that the new scheme preserves fourth order accuracy. The numerical applications contain the wave equation with and without a stiff nonlinear source term. This shows that different optimized schemes can be investigated for the solution of systems where physical terms with different behaviors exist.