Stiffness reduction of complex non-linear models and procedure to maintain solution quality

Wastewater treatment models consisting of large sets of non-linear ODE are usually stiff. Because stiff solvers cannot typically be applied, due to dynamic inputs, long computation times result. To limit the computational burden, model reduction, e.g. by linearization or by singular perturbation, has been applied on individual cases with good results, but there are no generalised methods to apply this in DAE systems. We therefore developed a method to improve the efficiency of a Diagonally Implicit Runge-Kutta (DIRK) DAE solver. The method consists of reducing the number of differential equations in the model by transforming some of them into algebraic equations, i.e. assuming instantaneous equilibrium. The Homotopy method is used to link the state variables to the large eigenvalues of the Jacobian (i.e. the fast dynamics). By solving these “stiff” state equations algebraically, important improvements in calculation time can be achieved. However, several practical issues remain. In particular, it is important to confirm that the reduction of the original model does not induce instability or unacceptable error. Control of instability is achieved by comparing the solution of five simulated steps computed with the reduced model to the solution of a single step of equivalent time computed with the full model. Since implicit Runge-Kutta methods show highly stable response, the full step can be considered as a converging estimate of the true solution, and thus a comparison point to detect instability. In case of unacceptable error (i.e. instability detected), the five simulated steps are rejected and the original model is used. An unacceptable error typically arises when a stiff state variable loses its stiffness after one step. This is common when states are influenced by non-linear equations in themselves, as apparent eigenvalues can drop dramatically when the state moves towards equilibrium. In such a case, the variation of the state variable away from equilibrium will be too large, missing potentially important events. The eigenvalue related to the state will pass from a very high eigenvalue to a much lower eigenvalue after a single simulated time step. To detect such a case, eigenvalues are recomputed after one time step and the number of “large” eigenvalues is compared before and after the time step. In case of a modification, the step is rejected and recalculated with the original model. Results have shown that improvements up to 45% in the number of function evaluations can be observed. Best improvements have been observed at more stringent error tolerances. Stability and error control have shown promising results, but the model reduction induces visible changes in intermediate values of stiff state variables. Caution must be used if results need high precision in all state variables.

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