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

作　　者：Jamal Alikhani Bahareh Shoghli Ujjal Kumar Bhowmik Arash Massoudieh Jamal Alikhani;Bahareh Shoghli;Ujjal Kumar Bhowmik;Arash Massoudieh(Department of Electrical Engineering and Computer Science, The Catholic University of America, Washington DC, USA;Department of Civil Engineering, The Catholic University of America, Washington DC, USA;Department of Civil Engineering, University of North Dakota, Grand Forks, ND, USA)

机构地区：[1]Department of Electrical Engineering and Computer Science, The Catholic University of America, Washington DC, USA [2]Department of Civil Engineering, The Catholic University of America, Washington DC, USA [2]Department of Civil Engineering, University of North Dakota, Grand Forks, ND, USA

出　　处：《American Journal of Computational Mathematics》2016年第4期298-312,共15页美国计算数学期刊（英文）

摘　　要：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.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.

关 键 词：Adaptive Time-Step Backward Differentiation Formula Activated Sludge Model Ordinary Differential Equation Stiffness Computation Time 

分 类 号：O17[理学—数学]