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

2016 ◽  
Vol 06 (04) ◽  
pp. 298-312 ◽  
Author(s):  
Jamal Alikhani ◽  
Bahareh Shoghli ◽  
Ujjal Kumar Bhowmik ◽  
Arash Massoudieh
Keyword(s):  
Differential Equations ◽  
Activated Sludge ◽  
Ordinary Differential Equations ◽  
Time Step ◽  
Stiff Ordinary Differential Equations ◽  
Adaptive Time Step ◽  
Adaptive Time ◽  
Activated Sludge Models

scholarly journals Stability Analysis of Singly Diagonally Implicit Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 211 ◽  
Author(s):  
Saufianim Jana Aksah ◽  
Zarina Ibrahim ◽  
Iskandar Mohd Zawawi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Triangular Matrix ◽  
Practical Significance ◽  
Computational Time ◽  
Step Size ◽  
Time Step ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

In this research, a singly diagonally implicit block backward differentiation formulas (SDIBBDF) for solving stiff ordinary differential equations (ODEs) is proposed. The formula reduced a fully implicit method to lower triangular matrix with equal diagonal elements which will results in only one evaluation of the Jacobian and one LU decomposition for each time step. For the SDIBBDF method to have practical significance in solving stiff problems, its stability region must at least cover almost the whole of the negative half plane. Step size restriction of the proposed method have to be considered in order to ensure stability of the method in computing numerical results. Efficiency of the SDIBBDF method in solving stiff ODEs is justified when it managed to outperform the existing methods for both accuracy and computational time.

Symplectic integrators with adaptive time step applied to runaway electron dynamics

2019 ◽  
Vol 81 (4) ◽  
pp. 1295-1309 ◽  
Author(s):  
Yanyan Shi ◽  
Yajuan Sun ◽  
Yang He ◽  
Hong Qin ◽  
Jian Liu
Keyword(s):  
Runaway Electron ◽  
Symplectic Integrators ◽  
Electron Dynamics ◽  
Time Step ◽  
Adaptive Time Step ◽  
Adaptive Time

scholarly journals A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 07 (01) ◽  
pp. 1350034 ◽  
Author(s):  
M. B. Suleiman ◽  
H. Musa ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Error Constant

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.

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