Discontinuities in Mathematical Modelling: Origin, Detection and Resolution

When modelling a chemical process, a modeller is usually required to handle a wide variations in time and/or length scales of its underlying differential equations by eliminating either the faster or slower dynamics. When compelled to deal with both and simultaneously simplify model structure, he/she is sometimes forced to make decisions that render the resulting model discontinuous. Discontinuities between adjacent regions, described by different equation sets, cause difficulties for ODE solvers. Two types exist for handling discontinuities in ODEs. Type I handles a discontinuity from the ODE solver side without paying any attention to the ODE model. This resolution to discontinuities suffer from underestimating the proper location of the discontinuity and thus results in solution errors. Type II discontinuity handlers resolve discontinuities at the model level by altering model structure or introducing bridging functions. This type of discontinuity handling has not been thoroughly explored in literature. I present a new hybrid (Type I and Type II) algorithm that eliminates integrator discontinuities through two steps. First, it determines the optimum switch point between two functions spanning adjacent or overlapping domains. The optimum switch point is determined by searching for a “jump point” that minimizes a discontinuity between adjacent/overlapping functions. Two resolution approaches exist. Approach I covers the entire overlap domain with an interpolating polynomial. Approach II relies on a moving vector to track a function trajectory during simulation run. Then, the discontinuity is resolved using an interpolating polynomial that joins the two discontinuous functions within a fraction of the overlap domain. The developed algorithm is successfully tested in models of a steady state chemical reactor exhibiting a bivariate discontinuity and a dynamic Pressure Swing Adsorption Unit exhibiting a univariate discontinuity in boundary conditions. Simulation results demonstrated a substantial increase in models' accuracy with a reduction in simulation runtime.