A fully adaptive reaction–diffusion integration scheme with applications to systems biology

Abstract

Adaptive integration schemes for ODE systems typically function by adjusting the time step size so as to keep the truncation error below some desired value. For adaptive integration of PDE systems involving coupled kinetic reaction and diffusion operations, truncation error arises not only from the individual propagators but also from their method of coupling. A common second-order accurate method for coupling operators is Strang’s method of operator splitting. We derive an expression for the truncation error resulting from Strang splitting reaction and diffusion operators for an arbitrary number of spatial dimensions, and demonstrate its use in adaptive time step algorithms. In addition, we present explanations of the second order implicit reaction and diffusion operators, and their individual error calculations used in our implementation of the scheme. Finally, using example simulations we discuss the use of this calculation for problems in systems biology.