TOWARDS UNDERSTANDING THE INFLUENCE OF GRADIENT RECONSTRUCTION METHODS ON UNSTRUCTURED FLOWSIMULATIONS

2017 ◽  
Vol 41 (2) ◽  
pp. 169-179 ◽  
Author(s):  
Fadi Mishriky ◽  
Paul Walsh
Keyword(s):  
Error Term ◽  
Computational Time ◽  
Test Case ◽  
Additional Time ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Gradient Reconstruction ◽  
Reconstruction Methods ◽  
Mesh Spacing ◽  
Cfd Applications

In this paper, the formal order of accuracy of three commonly used gradient reconstruction methods is derived. The analysis showed that the Green–Gauss cell based (GGCB) method is intrinsically inconsistent, due to the leading error term that is independent of the mesh spacing. On the other hand, the Green–Gauss node based (GGNB) and the Least Squares cell based (LSCB) methods achieved a minimum of 1st order accuracy regardless of the mesh geometric properties. Implications of the former results were practically tested on four CFD applications to show that in three out of four cases, the LSCB method achieved the highest order of accuracy. In terms of the computational expenses, the GGNB method consumed 9–34% additional time when compared to the fastest converging method in each test case. Both the GGCB and the LSCB methods consumed nearly the same computational time to reach convergence.

scholarly journals High-Order Hybrid WCNS-CPR Scheme for Shock Capturing of Conservation Laws

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jia Guo ◽  
Huajun Zhu ◽  
Zhen-Guo Yan ◽  
Lingyan Tang ◽  
Songhe Song
Keyword(s):  
Computational Cost ◽  
High Order ◽  
Shock Capturing ◽  
High Order Accuracy ◽  
Hybrid Technique ◽  
Hybrid Scheme ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Reconstruction Scheme ◽  
High Gradients

By introducing hybrid technique into high-order CPR (correction procedure via reconstruction) scheme, a novel hybrid WCNS-CPR scheme is developed for efficient supersonic simulations. Firstly, a shock detector based on nonlinear weights is used to identify grid cells with high gradients or discontinuities throughout the whole flow field. Then, WCNS (weighted compact nonlinear scheme) is adopted to capture shocks in these areas, while the smooth area is calculated by CPR. A strategy to treat the interfaces of the two schemes is developed, which maintains high-order accuracy. Convergent order of accuracy and shock-capturing ability are tested in several numerical experiments; the results of which show that this hybrid scheme achieves expected high-order accuracy and high resolution, is robust in shock capturing, and has less computational cost compared to the WCNS.

Comparison of Different Acceleration Techniques and Methods for Periodic Boundary Treatment in Unsteady Turbine Stage Flow Simulations

1999 ◽  
Author(s):  
Martin von Hoyningen-Huene ◽  
Alexander R. Jung
Keyword(s):  
Unsteady Flow ◽  
Numerical Method ◽  
Periodic Boundary ◽  
Computational Time ◽  
Test Case ◽  
The Real ◽  
Flow Simulations ◽  
Time Stepping ◽  
Acceleration Techniques ◽  
Pitch Ratio

This paper studies different acceleration techniques for unsteady flow calculations. The results are compared with a non-accelerated, fully-explicit solution in terms of time-averaged pressure distributions, the unsteady pressure and entropy in the frequency domain and the skin friction factor. The numerical method solves the unsteady three-dimensional Navier-Stokes equations via an explicit time-stepping procedure. The flow in the first stage of a modern industrial gas turbine is chosen as a test case. After a description of the numerical method used for the simulation, the test case is introduced. The comparison of the different numerical algorithms for explicit schemes is intended to ease the decision about which acceleration technique to use for calculations as far as accuracy and computational time are concerned. The convergence acceleration methods under consideration are, respectively, explicit time-stepping with implicit residual averaging, explicit time-consistent multigrid and implicit dual time stepping. The investigation and comparison of the different acceleration techniques are applicable to all explicit unsteady flow solvers. As another point of interest, the influence of the stage blade count ratio on the flow field is investigated. For this purpose, a simulation with a stage pitch ratio of unity is compared with a calculation using the real ratio of 78:80, which requires a more sophisticated method for periodic boundary condition treatment. This paper should help to decide whether it is vital from the turbine designer’s point of view to model the real pitch ratio in unsteady flow simulations in turbine stages.

scholarly journals On a fractional step-splitting scheme for the Cahn-Hilliard equation

2014 ◽  
Vol 31 (7) ◽  
pp. 1151-1168 ◽  
Author(s):  
A.A. Aderogba ◽  
M. Chapwanya ◽  
J.K. Djoko
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Computational Time ◽  
Fractional Step ◽  
Step Method ◽  
Fractional Step Method ◽  
Order Accuracy ◽  
Content Type ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.

scholarly journals High Order Godunov Type Multimesh Method for 3d Impact Problems of Elastoplastic Media

2021 ◽  
Keyword(s):  
Three Dimensional ◽  
Second Order ◽  
The Body ◽  
Godunov Scheme ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Sharp Interface Method ◽  
Impact Problems ◽  
Elastoplastic Media ◽  
Compact Stencil

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies under large displacements and deformations based on the multi mesh sharp interface method and modified Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The solutions of the Riemann problem for first and second order accuracy for compact stencil for an elastic medium in the case of an arbitrary stress state are obtained and presented, which are used at the “predictor” step of the Godunov scheme. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact stencil for moving Eulerian-Lagrangian grids. Modification is performed by converging the areas of influence of the differential and difference problems for the Riemann’s solver. The “corrector” step remains unchanged for both the first and second order accuracy schemes. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can change. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian-Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing obtain the parameters on the boundary and contact surfaces. The values of the underdetermined parameters in cell’s centers near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions by the Eulerian-Lagrangian and Lagrangian methods, as well as with experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

Multigranular Molecular Dynamics Simulations of Polymer Melts Using Multibody Algorithms

2005 ◽  
Author(s):  
Rudranarayan M. Mukherjee ◽  
Kurt S. Anderson ◽  
John Ziegler
Keyword(s):  
Molecular Dynamics ◽  
Polymer Melts ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Polymer Melt ◽  
Rigid Bodies ◽  
Computational Time ◽  
Integration Time ◽  
Test Case

In a multigranular approach for modeling molecular dynamics of polymer melts, different sections of the simulation box are modeled at different levels of detail viz. as particles, flexible bodies or rigid bodies. This approach eliminates high frequency localized motion while maintaining low frequency global conformational motion. This allows for longer integration time steps thus decreasing computational time. In this paper, we discuss our efforts to develop a consortium of dynamics algorithms capable of efficiently generating and solving the equations of motion at all three levels of modeling on a common software platform. A bead spring model of the polymer melt moving under the influence of truncated Lennard-Jones potential under periodic boundary conditions is pursued. Implementation issues and results from a test case consisting of 32 polymer chains of 16 beads each are presented. The paper also discusses the parallel implementation of this problem using MPI.

The Fourth Order of Accuracy Decomposition Scheme for Abstract Hyperbolic Equation

2008 ◽  
Vol 15 (1) ◽  
pp. 165-175
Author(s):  
Jemal Rogava ◽  
Mikheil Tsiklauri
Keyword(s):  
Approximate Solution ◽  
Hyperbolic Equation ◽  
Fourth Order ◽  
Operator Function ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Cosine Operator Function ◽  
Decomposition Scheme ◽  
Cosine Operator ◽  
Abstract Hyperbolic Equation

Abstract Using the rational splitting of a cosine operator-function, the fourth order accuracy decomposition scheme is constructed for hyperbolic equation when the principal operator is self-adjoint positively defined and is represented as a sum of two summands. Stability of the constructed scheme is shown and the error of an approximate solution is estimated.

Comparison of Different Acceleration Techniques and Methods for Periodic Boundary Treatment in Unsteady Turbine Stage Flow Simulations

1999 ◽  
Vol 122 (2) ◽  
pp. 234-246 ◽  
Author(s):  
Martin von Hoyningen-Huene ◽  
Alexander R. Jung
Keyword(s):  
Unsteady Flow ◽  
Numerical Method ◽  
Periodic Boundary ◽  
Computational Time ◽  
Test Case ◽  
The Real ◽  
Flow Simulations ◽  
Time Stepping ◽  
Acceleration Techniques ◽  
Pitch Ratio

This paper studies different acceleration techniques for unsteady flow calculations. The results are compared with a nonaccelerated, fully explicit solution in terms of time-averaged pressure distributions, the unsteady pressure and entropy in the frequency domain, and the skin friction factor. The numerical method solves the unsteady three-dimensional Navier–Stokes equations via an explicit time-stepping procedure. The flow in the first stage of a modern industrial gas turbine is chosen as a test case. After a description of the numerical method used for the simulation, the test case is introduced. The purpose of the comparison of the different numerical algorithms for explicit schemes is to facilitate the decision as to which acceleration technique should be used for calculations with regard to accuracy and computational time. The convergence acceleration methods under consideration are explicit time-stepping with implicit residual averaging, explicit time-consistent multigrid, and implicit dual time stepping. The investigation and comparison of the different acceleration techniques apply to all explicit unsteady flow solvers. This paper also examines the influence of the stage blade count ratio on the flowfield. For this purpose, a simulation with a stage pitch ratio of unity is compared with a calculation using the real ratio of 78:80, which requires a more sophisticated method for periodic boundary condition treatment. This paper should help to decide whether it is crucial from the turbine designer’s point of view to model the real pitch ratio in unsteady flow simulations in turbine stages. [S0889-504X(00)00702-9]

scholarly journals Study on Accuracy of the High-Resolution Schemes

2014 ◽  
Vol 6 ◽  
pp. 905053
Author(s):  
Yawen Tang ◽  
Bo Yu ◽  
Jianyu Xie ◽  
Jingfa Li ◽  
Peng Wang
Keyword(s):  
High Resolution ◽  
Numerical Solution ◽  
Richardson Extrapolation ◽  
Second Order ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Numerical Examples ◽  
High Resolution Schemes ◽  
Artificial Diffusion ◽  
Convection Dominated

The high-resolution (HR) schemes have been widely used as they can achieve the numerical solution without oscillation and artificial diffusion, especially for convection-dominated problems. However, there still have arguments about the order of accuracy of HR schemes, especially about the extreme value of the solution. In this paper, it is proved that any HR scheme designed in the NVD diagram has second-order accuracy when its combined segments totally locate in the BAIR region. In other words, it has been verified in our study that the segments, which have low-order accuracy when independently employed, have at least second-order accuracy when locate in BAIR region by analysis of two implementation methods of HR scheme and also a number of numerical examples. Meanwhile Richardson extrapolation has been used to estimate the order of accuracy of HR schemes which achieve the same conclusion.

scholarly journals Evaluation of phylogenetic reconstruction methods using bacterial whole genomes: a simulation based study

2018 ◽  
Vol 3 ◽  
pp. 33 ◽  
Author(s):  
John A. Lees ◽  
Michelle Kendall ◽  
Julian Parkhill ◽  
Caroline Colijn ◽  
Stephen D. Bentley ◽  
...  
Keyword(s):  
Maximum Likelihood ◽  
Phylogenetic Reconstruction ◽  
Simulated Data ◽  
Real Data ◽  
Tree Topology ◽  
Computational Time ◽  
Advantages And Disadvantages ◽  
Phylogenetic Reconstructions ◽  
Reconstruction Methods ◽  
True Tree

Background: Phylogenetic reconstruction is a necessary first step in many analyses which use whole genome sequence data from bacterial populations. There are many available methods to infer phylogenies, and these have various advantages and disadvantages, but few unbiased comparisons of the range of approaches have been made. Methods: We simulated data from a defined 'true tree' using a realistic evolutionary model. We  built phylogenies from this data using a range of methods, and compared reconstructed trees to the true tree using two measures, noting the computational time needed for different phylogenetic reconstructions. We also used real data from Streptococcus pneumoniae alignments to compare individual core gene trees to a core genome tree. Results: We found that, as expected, maximum likelihood trees from good quality alignments were the most accurate, but also the most computationally intensive. Using less accurate phylogenetic reconstruction methods, we were able to obtain results of comparable accuracy; we found that approximate results can rapidly be obtained using genetic distance based methods. In real data we found that highly conserved core genes, such as those involved in translation, gave an inaccurate tree topology, whereas genes involved in recombination events gave inaccurate branch lengths. We also show a tree-of-trees, relating the results of different phylogenetic reconstructions to each other. Conclusions: We recommend three approaches, depending on requirements for accuracy and computational time. For the most accurate tree, use of either RAxML or IQ-TREE with an alignment of variable sites produced by mapping to a reference genome is best. Quicker approaches that do not perform full maximum likelihood optimisation may be useful for many analyses requiring a phylogeny, as generating a high quality input alignment is likely to be the major limiting factor of accurate tree topology.  We have publicly released our simulated data and code to enable further comparisons.

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