scholarly journals Three-step Predictor-Corrector Finite Element Schemes for Consolidation Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Mina Torabi ◽  
Manuel Pastor ◽  
Miguel Martín Stickle
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Spectral Element ◽  
Euler Method ◽  
One Dimensional ◽  
Time Integration Method ◽  
The One ◽  
Predictor Corrector ◽  
Finite Element Schemes ◽  
First Time

An accurate, stable, and efficient three-step predictor-corrector time integration method is considered, for the first time, to obtain numerical solution for the one-dimensional consolidation equation within a finite and spectral element framework. Theoretical order of accuracy and stability conditions are provided. The three-step predictor-corrector time integration method is third-order accurate and shows a larger stability region than the forward Euler method when applied to the one-dimensional consolidation equation. Furthermore, numerical results are in agreement with analytical solutions previously derived by the authors.

scholarly journals Optimization of one-parameter family of integration formulae for solving stiff chemical-kinetic ODEs

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Youhi Morii ◽  
Eiji Shima
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Chemical Kinetic ◽  
Parameter Family ◽  
Minimum Error ◽  
Time Stepping ◽  
Time Integration Method ◽  
Dual Time Stepping ◽  
Stiff Odes ◽  
The One

AbstractA fast and robust Jacobian-free time-integration method—called Minimum-error Adaptation of a Chemical-Kinetic ODE Solver (MACKS)—for solving stiff ODEs pertaining to chemical-kinetics is proposed herein. The MACKS formulation is based on optimization of the one-parameter family of integration formulae coupled with a dual time-stepping method to facilitate error minimization. The proposed method demonstrates higher accuracy compared to the method—Extended Robustness-enhanced numerical algorithm (ERENA)—previously proposed by the authors. Additionally, when this method is employed in homogeneous-ignition simulations, it facilitates realization of faster performance compared to CVODE.

scholarly journals Composite Backward Differentiation Formula for the Bidomain Equations

2020 ◽  
Vol 11 ◽  
Author(s):  
Xindan Gao ◽  
Craig S. Henriquez ◽  
Wenjun Ying
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Euler Method ◽  
Implicit Methods ◽  
Differentiation Formula ◽  
Bidomain Equations ◽  
Time Integration Method ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Backward Euler

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.

Development of a Log-Time Integration Method for Reactive Flows

2012 ◽  
Author(s):  
Jose Escobar ◽  
Ismail Celik ◽  
Donald Ferguson
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Chemical Species ◽  
Transport Equations ◽  
Computational Time ◽  
Species Transport ◽  
One Dimensional ◽  
Time Integration Method ◽  
Runge Kutta Method ◽  
Runge Kutta

In reactive flow simulations integration of the stiff species transport equations consumes most of the computational time. Another important aspect of combustion simulation is the need to simulate at least tens of species in order to accurately predict emissions and the related combustion dynamics. Small time scales and systems with tens of species lead to very high computational costs. Classic integration methods such as Euler method are restricted by the smallest characteristic time scale, and explicit Runge-Kutta methods require intermediate predictor corrector steps which make the problem computationally expensive. On the other hand, implicit methods are also computationally expensive due the calculation of the Jacobian. This work presents a strategy to significantly reduce computational time for integration of species transport equations using a new explicit integration scheme called Log-Time Integration Method (LTIM). LTIM is fairly robust and can compete with methods such as the 5th order Runge-Kutta method. Results showed that LTIM applied to the solution of a zero dimensional reactive system which consists of 4 chemical species obtains the solution around 4 times faster than 5th order Runge-Kutta method. LTIM was also applied to the solution of a one dimensional methane-air flame. The chemical reactions were modeled using a reduced chemical mechanism ARM9 that consists of 9 chemical species and 5 global reactions. The solution was carried out for 9 species transport equations along with the energy equation. Governing equations were decoupled into flow and chemical parts and were solved separately using a split formulation. Thermodynamic properties were obtained using NASA format polynomials and transport properties using kinetic-theory formulation. It is shown that the new one dimensional flame code is able to calculate the adiabatic flame temperature of the system and corresponding flame speed for the methane-air flame thus validating its robustness and accuracy.

scholarly journals An Improved Time Integration Method Based on Galerkin Weak Form with Controllable Numerical Dissipation

2021 ◽  
Vol 11 (4) ◽  
pp. 1932
Author(s):  
Weixuan Wang ◽  
Qinyan Xing ◽  
Qinghao Yang
Keyword(s):  
High Frequency ◽  
Integration Method ◽  
Time Integration ◽  
Low Frequency ◽  
Weak Form ◽  
High Accuracy ◽  
Numerical Dissipation ◽  
Frequency Range ◽  
Time Integration Method ◽  
Numerical Damping

Based on the newly proposed generalized Galerkin weak form (GGW) method, a two-step time integration method with controllable numerical dissipation is presented. In the first sub-step, the GGW method is used, and in the second sub-step, a new parameter is introduced by using the idea of a trapezoidal integral. According to the numerical analysis, it can be concluded that this method is unconditionally stable and its numerical damping is controllable with the change in introduced parameters. Compared with the GGW method, this two-step scheme avoids the fast numerical dissipation in a low-frequency range. To highlight the performance of the proposed method, some numerical problems are presented and illustrated which show that this method possesses superior accuracy, stability and efficiency compared with conventional trapezoidal rule, the Wilson method, and the Bathe method. High accuracy in a low-frequency range and controllable numerical dissipation in a high-frequency range are both the merits of the method.

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