scholarly journals Approximate Factorization Method Using Alternating Cell Direction Implicit Method: Comparison of Convergence Characteristics Using Basic Model Equations

2021 ◽  
Vol 48 (3) ◽  
Author(s):  
Ali Ruhs¸en C¸ ETE ◽  
Keyword(s):  
Unstructured Grids ◽  
Factorization Method ◽  
Unstructured Meshes ◽  
Implicit Method ◽  
Approximate Factorization ◽  
Time Step ◽  
Fully Implicit ◽  
Implicit Schemes ◽  
Iteration Methods ◽  
Implicit Iteration

In this paper, a fast implicit iteration scheme called the alternating cell directions implicit (ACDI) method is combined with the approximate factorization scheme. The use of fast implicit iteration methods with unstructured grids is hardly. The proposed method allows fast implicit formulations to be used in unstructured meshes, revealing the advantages of fast implicit schemes in unstructured meshes. Fast implicit schemes used in structured meshes have evolved considerably and are much more accurate and robust, and are faster than explicit schemes. It is a crucial novel development that such developed schemes can be applied to unstructured schemes. In steady incompressible potential flow, the convergence character of the scheme is compared with the Runge-Kutta order 4 (RK4), Laasonen, point Gauss–Seidel iteration, old version ACDI, and line Gauss–Seidel iteration methods. The scheme behaves like an approximation of the fully implicit method (Laasonen) up to an optimum pseudo-time-step size. This is a highly anticipated result because the approximate factorization method is an approach to a fully implicit formulation. The results of the numerical study are compared with other fast implicit methods (e.g., the point and line Gauss–Seidel methods), the RK4 method, which is an explicit scheme, and the Laasonen method, which is a fully implicit scheme. The study increased the accuracy of the ACDI method. Thus, the new ACDI method is faster in unstructured grids than other methods and can be used for any mesh construction.

An Implicit Algorithm for Stator-Rotor Interaction Analysis

1996 ◽  
Author(s):  
V. Michelassi ◽  
P. Adami ◽  
F. Martelli
Keyword(s):  
Interaction Analysis ◽  
Experimental Testing ◽  
Three Dimensional ◽  
Factorization Method ◽  
Steady Flows ◽  
Turbine Stage ◽  
Approximate Factorization ◽  
Time Step ◽  
Physical Time ◽  
Speed And Accuracy

A simple time-accurate algorithm is presented for the computation of the unsteady stator-rotor interaction. The algorithm is based on the scalar approximate factorization method originally developed for the computation of complex three-dimensional steady flows. The method introduces a physical time step, used to march in time, and a numerical time step to iterate in between physical time steps. The method is formulated so as to take full advantage of the implicit formulation and provide an implicit treatment of the unsteady terms. A set of preliminary tests on a turbine stage, still in the experimental testing phase, proved the speed and accuracy of the method which was able to capture the essential features of a transonic stage.

A Fully Implicit Finite Volume Lattice Boltzmann Method for Turbulent Flows

2017 ◽  
Vol 22 (2) ◽  
pp. 393-421
Author(s):  
Fatih Çevik ◽  
Kahraman Albayrak
Keyword(s):  
Lattice Boltzmann Method ◽  
Turbulent Flows ◽  
Finite Volume ◽  
Lattice Boltzmann ◽  
Factorization Method ◽  
Implicit Methods ◽  
Approximate Factorization ◽  
Fully Implicit ◽  
Almost All ◽  
Boltzmann Method

AbstractAlmost all schemes existed in the literature to solve the Lattice Boltzmann Equation likestream & collide, finite difference, finite element, finite volumeschemes are explicit. However, it is known fact that implicit methods utilizes better stability and faster convergence compared to the explicit methods. In this paper, a method named herein as Implicit Finite Volume Lattice Boltzmann Method (IFVLBM) for incompressible laminar and turbulent flows is proposed and it is applied to some 2D benchmark test cases given in the literature.Alternating Direction Implicit, an approximate factorization method is used to solve the obtained algebraic system. The proposed method presents a very good agreement for all the validation cases with the literature data. The proposed method shows good stability characteristics, the CFL number is eased. IFVLBM has about 2 times faster convergence rate compared with Implicit-Explicit Runge Kutta method even though it possesses a computational burden from the solution of algebraic systems of equations.

Reservoir Simulation Using an Adaptive Implicit Method

1983 ◽  
Vol 23 (05) ◽  
pp. 759-768 ◽  
Author(s):  
G.W. Thomas ◽  
D.H. Thurnau
Keyword(s):  
Reservoir Simulation ◽  
Computing Time ◽  
Substantial Reduction ◽  
Implicit Method ◽  
Time Step ◽  
Mathematical Procedure ◽  
Fully Implicit ◽  
Rapid Changes ◽  
Grid Block ◽  
And Storage

Abstract This paper deals with a new implicit method for reservoir simulation. Rather than provide a fixed degree of implicitness in every grid block at every time step or iteration, the adaptive implicit method operates with different levels of implicitness in adjacent grid blocks. These levels shift in space and tithe as needed to maintain stability. Shifting is accomplished automatically without user intervention. The technique can he applied to any simulation problem involving N unknowns. The advantage is substantial reduction in computing time and storage requirements compared to fully implicit formulations, while still yielding unconditionally stable solutions. The mathematical procedure involves labeling the impplicit/explicit mix of unknowns and then composing the matrix problem. The latter is reducible as long, as there is one or more unknowns to be computed explicitly; consequently, appropriate operations are performed to put it in reduced form. This leads to matrix equations of lower order than the original problem that are solved at less cost. We demonstrate each step of the mathematical procedure with an example. procedure with an example. Finally, applications to a three-phase coning problem and a three-dimensional (3D) Cartesian problem are presented. By using special displays we demonstrate the presented. By using special displays we demonstrate the degrees of implicitness in each cell and how they shift in space and time during simulation. We also present information regarding the savings in computer time and storage compared with a fixed, fully implicit procedure. Introduction With the growth of reservoir simulation technology, the tendency has been to develop simulators that offer implicit or near-fully implicit capabilities. Implicit calculation are often necessary to maintain stability when one or more of the computed variables (pressure. saturation, temperature, composition, etc.) undergoes large surges over a time step. An advantage one accrues from a highly implicit simulator, besides stability, is that large timestep sizes can be tolerated. However, this advantage is offset by larger time-truncation errors, and substantially higher processor times and storage requirements. Moreover, most commercial reservoir simulators provide only a fixed level of implicitness in every grid block at every timestep. Typically, however, only a small fraction of the total grid blocks in a reservoir model undergo rapid changes in the computed variables to justify high levels of implicitness. For example, in a well undergoing gas and water coning, rapid changes in saturation and pressure occur only in the grid blocks in the near-well region, while farther away the changes are more subdued. A fully implicit model similar to that discussed in Ref. 1 adequately handles such a situation with guaranteed stability. But obviously, some overkill is involved in those cells where the changes are modest. On the other hand, an implicit pressure explicit saturation (IMPES) treatment of such a problem can result in serious underkill and instability in the blocks near the wellbore, unless the simulator is especially modified. The overkill problem cited above is substantially worse in large global reservoir simulations where one or more intervening grid blocks separate the well blocks. SPEJ P. 759

scholarly journals A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1639
Author(s):  
Abdelkrim Aharmouch ◽  
Brahim Amaziane ◽  
Mustapha El Ossmani ◽  
Khadija Talali
Keyword(s):  
Finite Volume ◽  
Coastal Aquifers ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Mass Conservation ◽  
Coupled System ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Volume Scheme ◽  
Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

Tests of the Stability and Time-Step Sensitivity of Semi-Implicit Reservoir Stimulation Techniques

1972 ◽  
Vol 12 (03) ◽  
pp. 253-266 ◽  
Author(s):  
James S. Nolen ◽  
D.W. Berry
Keyword(s):  
Finite Difference ◽  
Difference Equations ◽  
Implicit Method ◽  
Significant Advance ◽  
Relative Permeabilities ◽  
Time Step ◽  
Finite Difference Equations ◽  
Nonlinear Form ◽  
Implicit Finite Difference ◽  
Excellent Stability

Abstract A reservoir simulation technique that employs semi-implicit approximations to relative permeabilities exhibits excellent stability and permeabilities exhibits excellent stability and convergence characteristics when applied to water- or gas-coning problems. Recent workers in this area have made a simplifying assumption in order to linearize the flow terms of the semi-implicit finite-difference equations. This paper describes a method of solving efficiently paper describes a method of solving efficiently the nonlinear form of the equations and demonstrates that time-step sensitivity is reduced by iterating on the nonlinear terms. In addition, it addresses the problem of allocating a well's production among multiple grid blocks. Example problems include both water-coning and gas-percolation applications. Introduction Multiphase reservoir simulators traditionally have employed finite-difference approximations in which relative permeabilities are evaluated explicitly at the beginning of each time step. Simulators of this type are capable of handling many reservoir studies in a perfectly satisfactory fashion, but they are incapable of solving economically problems characterized by high flow velocities. Included in this category are the studies of such phenomena as well coning and gas percolation. The difficulty in such problems is a stability limitation imposed by the use of explicit relative permeabilities. In an attempt to overcome this permeabilities. In an attempt to overcome this limitation, Blair and Weinaug developed a simulator that employed implicitly evaluated relative permeabilities. The increased stability of their permeabilities. The increased stability of their equations allowed the use of time steps much larger than previously possible, but this was counteracted by an increase in the computational work per time step and an increased difficulty in the iterative solution of the difference equations. While the net result was a significant advance in the solution of coning problems, improvements still were needed to increase the dependability and decrease the cost of obtaining solutions for such problems. More recently, two papers were published describing a method that employs semi-implicit relative permeabilities. This method is greatly superior to the fully implicit method, both in computational effort and maximum time-step size. In developing this method, the previous workers made a simplifying assumption to obtain linear finite-difference equations. We have developed a reservoir simulator based on the nonlinear form of the semi-implicit finite-difference equations. This paper describes the techniques used in the simulator and presents the results of some tests conducted with it. These include time-step sensitivity studies and tests of alternate production allocation methods. Some of these tests compare the nonlinear form of the semi-implicit method with the linear form. SPEJ P. 253

scholarly journals Thermal modeling of hydrogen storage by absorption in a magnesium hydrides tank

2014 ◽  
Vol 5 ◽  
pp. A21 ◽  
Author(s):  
Karim Lahmer ◽  
Rachid Bessaïh ◽  
Angel Scipioni ◽  
Mohammed El Ganaoui
Keyword(s):  
Hydrogen Absorption ◽  
Numerical Scheme ◽  
Computational Time ◽  
Governing Equations ◽  
Time Step ◽  
Kinetic Reaction ◽  
Fully Implicit ◽  
Temporal Profiles ◽  
Tank Geometry ◽  
Reaction Equations

This paper summarizes numerical results of hydrogen absorption simulated in an axisymmetric tank geometry containing magnesium hydride heated to 300 °C and at moderate storage pressure 1 MPa. The governing equations are solved with a fully implicit finite volume numerical scheme used by a commercial software FLUENT. The effect of the different kinetic reaction equations modeling hydrogen absorption was studied by the introduction of a specific subroutine at each time step in order to consider which one will provide results close to available experimental results. Spatial and temporal profiles of temperature and concentration in hydride bed are plotted. Results show that suitable method for our two-dimensional study is a CV-2D technique because it generates the smallest error especially during the beginning of the reaction. Also, its computational time is the shortest one compared to the other methods.

Sign in / Sign up

Export Citation Format

Share Document