There are two parts of my thesis. The first part is about muilti-step Krylov IIF-WENO methods. When we are dealing with time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms, implicit integration factor (IIF) methods are often a good choice as a class of efficient “exactly linear part” time discretization method. In complex systems (e.g. advection-diffusion-reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear non-stiff terms and nonlinear stiff terms are often mixed together. To discretize the hyperbolic part in ADR systems, high order weighted essentially non-oscillatory (WENO) methods are often used. There are two open problems related to IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. We solve these two problems by developing new muilti-step Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. Large time step size computations are obtained.
The second part is about single-step Krylov IIF-WENO methods. So far, the IIF methods developed in the literature are multi-step methods. Hence, we develop single-step Krylov IIF-WENO methods for solving stiff advection-diffusion-reaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of both the single-step and multi-step schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of these two schemes.