Many processes found in science and engineering are governed by dynamical systems described by ordinary and partial differential equations. These systems are often complex and do not admit closed form solutions. Therefore, numerical methods called time integrators are required for finding solutions. In this thesis we study the effeciency of various integrators for solving three partial differential equations: the heat equation, the viscous Burgers’ equation, and Stokes equations (specifically the time integration of velocity field produced by method of regularized Stokeslets). A time integrator’s efficiency is quantified by analyzing the amount of computational time required to approximate the solution at a given accuracy. This thesis has three primary components. First, we will discuss implicit and explicit integrators for solving linear PDEs. Second, we will present implicit-explicit and exponential integrators for solving semi-linear PDEs. Lastly, we propose to use a semi-linear time integrator for solving initial value problems arising from the Stokes equations. We find that a semi-implicit integrator can take larger timesteps when modeling stiff springs.