Analysis and applications of combined numerical methods for systems of differential equations

The goal of this dissertation is to study combined numerical methods for solving systems of ordinary differential equations. We shall establish a general framework for such methods and develop a complete convergence theory for both combined linear multistep methods and combined Runge-Kutta methods and seek efficient new implicit-explicit Runge-Kutta methods for solving stiff problems. The performance of the new methods will be investigated by applying them to integrate a dynamical system of numerical weather prediction.; The framework of combined numerical methods is generalized from many useful semi-implicit time differencing schemes and the ADI method that have been used for the time integration of spatially discretized time-dependent partial differential equations. For example, the most common approach to solve a convection-diffusion problem numerically is to first discretize the equations in space by finite difference, finite element, or spectral methods, then use an implicit scheme for the diffusion terms and an explicit scheme for the convection terms to advance in time. Such time advancing schemes can reduce the restrictive stability conditions on time steps caused by the stiffness of the problem, yet the resulting implicit discrete problems can be solved without much effort.; In this work, we shall discuss how to construct combined numerical methods by combining different linear multistep methods or Runge-Kutta methods. A convergence theory will be developed to prove such combinations are convergent if the methods being combined are convergent. The order conditions of combined methods will be derived. We shall also take a close look at the local truncation error of such methods.; A linear stability analysis is conducted to explore all the possible implicit-explicit Runge-Kutta methods of orders from one to three with the minimal number of stages. Some methods permitting large time steps for solving stiff systems of differential equations are discovered. Our analysis shows that implicit-explicit Runge-Kutta methods have generally bigger stability regions than implicit-explicit linear multistep methods of Ascher et al. (2).; The new methods identified by the stability analysis are tested using the spectral anelastic model developed by Fulton (15). The numerical results agree with the convergence and stability analysis. It is demonstrated that a higher-order time differencing improves the overall efficiency when the equations are discretized in space by higher-order methods such as spectral approximations. It is also shown that the third order implicit-explicit Runge-Kutta method has better stability properties than third-order implicit-explicit linear multistep methods for the spectral anelastic model.