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Implicit Methods
Published in Niket S. Kaisare, Computational Techniques for Process Simulation and Analysis Using MATLAB®, 2017
One of the most popular numerical methods for solving stiff ODEs and DAEs is Gear’s method, which uses BDF of varying orders, from first to fifth order. BDF methods are popular for solving highly stiff problems because they have good stability (up to fifth-order), reasonably high accuracy, and capability to solve differential equations with algebraic constraints. In case of DAE problems, the differential equations retain the entire formula from Table 8.3, whereas the left-hand side is zero for algebraic equations.
Rutting prediction of nanomaterial modified asphalt concrete using FEM simulation
Published in Sandra Erkens, Xueyan Liu, Kumar Anupam, Yiqiu Tan, Functional Pavement Design, 2016
The proposed viscoelastic viscoplastic damage model consists of stiff differential equations and it is difficult to obtain an analytical solution. Therefore, the Backward Difference Formulas (BDF) are adopted to solve the constitutive equations. BDF is a family of implicit methods for the numerical integration of ordinary differential equations, especially for stiff differential equations.
An improved stiff-ODE solving framework for reacting flow simulations with detailed chemistry in OpenFOAM
Published in Combustion Theory and Modelling, 2023
Kun Wu, Yuting Jiang, Zhijie Huo, Di Cheng, Xuejun Fan
To be specific, in finite-rate chemistry approach, the computational cost associated with the chemistry solution originated from three major aspects, including the ODE-integration, Jacobian evaluation, and linear system solution. The numerical algorithm of an ODE integrator decisively determines the order of accuracy and computational efficiency. Owing to the better efficiency and stability, chemistry systems are commonly solved using implicit integration methods, among which the multi-step, backward differentiation formulation (BDF) method [12,13] is the most widely used one for combustion simulations [14]. Despite its high-order of accuracy, BDF method suffers from the re-initialisation problem and is prone to accumulation error, since it always starts with low-order approximation and builds up higher order successively [15,16]. To alleviate the concentration of computational cost at the beginning of every iteration, Imren and Haworth [15] resorted to an extrapolation-based method namely Seulex. The Seulex algorithm applies semi-implicit Euler method on the recursively partitioned sub-intervals and successively improves the solution accuracy through high-order projection [17]. As a one-step method, Seulex requires less information to reinitialise integration and achieves better performance than BDF method [15].
High-Order Accurate Solutions of the Point Kinetics Equations with the Spectral Deferred Correction Method
Published in Nuclear Science and Engineering, 2023
The most popular high-order methods in the neutronics community are the -method,4–6 the backward differentiation formula7 (BDF), and the Runge-Kutta (RK) methods.8 A thorough review of these methods can be found in Ref. 9. The -method can have at most second-order accuracy. When increasing the order of accuracy through the BDF or RK methods, the implementation becomes more complicated and the fundamental limits in the stability or efficiency of these methods also arise. For example, there exists no stable BDF method of order greater than six, and high-order implicit RK methods can be extremely computationally expensive.10
Aerodynamic Tailoring of Structures Using Computational Fluid Dynamics
Published in Structural Engineering International, 2019
Fei Ding, Ahsan Kareem, Jiawei Wan
In addition to this, an alternate way for preserving the order of the time-accuracy of the preferred time-integrator that satisfies the geometric conservation law in addition to the method given in Refs. [55] is presented. The presented approach not only applies to the BDF-2 considered here but also to other time-integration schemes like Runge–Kutta. The accuracy and convergence results of a specified time-integrator in solving the fluid subsystem can be analyzed using the existing theories related DAE cases. It is also worth mentioning that the BDF-2 is chosen here since it has the same order of local accuracy and global convergence involving ODE and DAE cases of Index 1 or 2.