TITLE:
An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models
AUTHORS:
Jamal Alikhani, Bahareh Shoghli, Ujjal Kumar Bhowmik, Arash Massoudieh
KEYWORDS:
Adaptive Time-Step, Backward Differentiation Formula, Activated Sludge Model, Ordinary Differential Equation, Stiffness, Computation Time
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.6 No.4,
November
11,
2016
ABSTRACT: A backward differentiation formula (BDF) has been shown to be an
effective way to solve a system of ordinary differential equations (ODEs) that
have some degree of stiffness. However, sometimes, due to high-frequency
variations in the external time series of boundary conditions, a small
time-step is required to solve the ODE system throughout the entire simulation
period, which can lead to a high computational cost, slower response, and need
for more memory resources. One possible strategy to overcome this problem is to
dynamically adjust the time-step with respect to the system’s stiffness.
Therefore, small time-steps can be applied when needed, and larger time-steps
can be used when allowable. This paper presents a new algorithm for adjusting
the dynamic time-step based on a BDF discretization method. The parameters used
to dynamically adjust the size of the time-step can be optimally specified to
result in a minimum computation time and reasonable accuracy for a particular
case of ODEs. The proposed algorithm was applied to solve the system of ODEs
obtained from an activated sludge model (ASM) for biological wastewater
treatment processes. The algorithm was tested for various solver parameters,
and the optimum set of three adjustable parameters that represented minimum
computation time was identified. In addition, the accuracy of the algorithm was
evaluated for various sets of solver parameters.