This paper focuses on one way a linearized representation of a nonlinear economic model can be used to obtain arbitrarily accurate solutions to simulations. The key is a method for translating a simulation problem directly to a so-called initial value problem. Since many different methods for solving initial value problems are known and well understood, and since each one converts to an algorithm for solving simulation problems, this insight greatly expands the computational tool kit for conducting simulations. This paper contains a survey of the theoretical results guaranteeing convergence and forming the basis for extrapolations of two important methods for solving initial value problems. Theoretical considerations suggest that the faster rate of convergence of one of these methods (the modified midpoint method) is likely to cause it to dominate the other (Euler's method) in many situations faced by applied general equilibrium modellers. The other main points of the paper are: (i) to emphasize that linearized (symbolic) representations of models lead naturally to efficient algorithms which can be used to compute solutions having any desired degree of precision; and (ii) to suggest that such accurate methods (rather than Johansen's method) should be the default when solving models (especially applied general equilibrium models) represented in linearized form.