Accuracy of Parameter Sensitivities of DAE Systems using Finite Difference Methods

This work evaluates the accuracy of Finite Difference (FD) methods when computing parameter sensitivities of badly-scaled DAE systems. Due to their simple implementation, FD methods are commonly favoured especially when the underlying mathematical model is a hard-coded sophisticated simulator. Nevertheless, FD methods may impose serious numerical problems even if FD step sizes and solver tolerances w.r.t. the order of the FD scheme are ideally selected. Judging the precision of the resulting parameter sensitivities is practically difficult. With the availability of powerful Automatic Differentiation (AD) tools for equation-based simulation languages like ADModelica, there is a new possibility to examine step sizes of various FD schemes, solver tolerances and the resulting precision for realistic large scale examples. This can be done by comparing numerical parameter sensitivities with highly precise analytical solutions using direct integration of sensitivity equation systems generated by AD techniques. It is shown with a realistically sized example that FD methods are actually more critical than usually assumed.