Runge-Kutta integrators for fast and accurate solutions in GEMPACK

In GEMPACK, models are always solved as initial value problems(IVP)using the linearized form of the levels equations. While this allows the user to solve each step of the IVP efficiently, the overall accuracy and speed is determined by the integration scheme and the number of integration steps. Up to GEMPACK 12.1, only the Euler, leapfrog midpoint and Gragg’s method were available as well as their 2 and 3 point Richardson extrapolations. While Euler provides excellent stability it is very costly to obtain accurate solutions. In contrast the latter two integrators allow for faster convergence but oftentimes suffer from instabilities. In the current beta version of GEMPACK we address this issue by introducing explicit and embedded Runge Kutta (RK) integrators as an alternative. Our focus in this work is on the embedded RK methods. Using the embedded RK methods we developed a new adaptive step size algorithm that is designed to overcome problem common to CGE models. Such problems include asymptotes in the levels variables as well as coping with the different scales on which the results can vary. Our algorithm provides rapid convergence towards the true solution as well as increased robustness exceeding that of Eulers method. In addition, the new algorithm allows us to provide users with a component-by-component global error estimate. In all our tests we have found that the error estimates appeared to be upper bounds of the true error. Furthermore, this compenent-by-component error estimates are an excellent debugging tool when developing or extending a CGE model.In all but the simplest test cases, we have found that using adaptive step size embedded RK methods provided solutions at least one order of magnitude closer to the true solution in less than half the time to solution required by the old integration schemes.