An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

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.

Investigating Effects of Heterogeneity and Fracture Distribution on Two-Phase Flow in Fractured Reservoir with adaptive time strategy

“…An appropriate Δt enables a low cost iteration as well as a good convergence performance. The optimal option is to set a variable Δt which can be updated dynamically during the iteration (Alikhani et al 2016;Shepherd et al 2019). The procedure of adaptive time step control is shown in Algorithm 1.…”

Application of Bayesian Approach in the Parameter Estimation of Continuous Lumping Kinetic Model of Hydrocracking Process

“…Therefore, an efficient numerical algorithm [33] should be selected to reduce the overall computation time of each simulation run.…”

Testing and improving a non-conventional unconditionally positive finite difference method

“…Since the rate of change of the T functions and even the degree of stiffness during the course of the simulation can vary, using a fixed but small time step implies a heavy computational burden. A dynamically adaptive time step algorithm can automatically adjust the time step according to the accuracy requirements and the degree of stiffness, resulting in a more optimal use of computational resources [3]. Successful adaptive methods may lead to substantial savings in computational work for a given accuracy [4].…”