scholarly journals Improving Initial Guess for the Iterative Solution of Linear Equation Systems in Incompressible Flow

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 119
Author(s):  
Shuai Ye ◽  
Yufei Lin ◽  
Liyang Xu ◽  
Jiaming Wu
Keyword(s):  
Linear Equation ◽  
Stokes Equations ◽  
Numerical Test ◽  
Weighted Average ◽  
Iterative Solution ◽  
Equation System ◽  
Oscillatory Solution ◽  
Initial Guess ◽  
Time Step ◽  
Linear Equation Systems

The pressure equation, generated while solving the incompressible Navier–Stokes equations with the segregated iterative algorithm such as PISO, produces a series of linear equation systems as the time step advances. In this paper, we target at accelerating the iterative solution of these linear systems by improving their initial guesses. We propose a weighted group extrapolation method to obtain a superior initial guess instead of a general one, the solution of the previous linear equation system. In this method, the previous solutions that are used to extrapolate the predicted solutions are carefully organized to address the oscillatory solution on each grid. The proposed method uses a weighted average of the predicted solutions as the new initial guess to avoid over extrapolating. Three numerical test results show that the proposed method can accelerate the iterative solution of most linear equation systems and reduce the simulation time up to 61.3%.

Numerical solution methods

2018 ◽  
Author(s):  
Hans Fehr ◽  
Fabian Kindermann
Keyword(s):  
Numerical Solution ◽  
Linear Equation ◽  
Equation System ◽  
Economic Problem ◽  
Solution Technique ◽  
Linear Equation System ◽  
Fortran Code ◽  
Numerical Solution Methods ◽  
Solution Methods ◽  
Linear Equation Systems

In this chapter we develop simple methods for solving numerical problems. We start with linear equation systems, continue with nonlinear equations and finally talk about optimization, interpolation, and integration methods. Each section starts with a motivating example from economics before we discuss some of the theory and intuition behind the numerical solution method. Finally, we present some Fortran code that applies the solution technique to the economic problem. This section mainly addresses the issue of solving linear equation systems. As a linear equation system is usually defined by a matrix equation, we first have to talk about how to work with matrices and vectors in Fortran. After that, we will present some linear equation system solving techniques.

scholarly journals Pengembangan LKPD berbasis PBL untuk Meningkatkan Kemampuan Pemecahan Masalah Siswa Kelas VIII pada Materi Sistem Persamaan Linear Dua Variabel

2021 ◽  
Vol 1 (4) ◽  
pp. 631-640
Author(s):  
Aflahul Ma'wa ◽  
Hapipi Hapipi ◽  
Muhammad Turmuzi ◽  
Syahrul Azmi
Keyword(s):  
Problem Solving ◽  
Linear Equation ◽  
Teaching And Learning ◽  
Problem Based Learning ◽  
Equation System ◽  
Design Development ◽  
School Year ◽  
Linear Equation System ◽  
Class Viii ◽  
Linear Equation Systems

The purpose of this study was to develop student worksheets based on PBL (Problem Based Learning) to improve the problem-solving abilities of class VIII students at MTs Hikmatusysyarief NW Salut Narmada for the 2020-2021 school year on two-variable linear equation systems. PBL-based LKPD (Problem Based Learning) is an LKPD that is arranged based on the phases of the PBL learning model. The development of this product aims to overcome the weaknesses in the teaching and learning process and overcome the weaknesses of students' problem-solving abilities. The research methodology used in developing PBL-based LKPD is 4D (Define, Design, Development, and Dissemination). Data collection techniques using interview guidelines and questionnaires. The result of the research is a product in the form of LKPD based on PBL (Problem Based Learning) on ​​the material of a two-variable linear equation system. The product has been declared valid with very good criteria by two validators. Based on the results of a limited trial on the use of the developed product, it is known that the PBL-based LKPD can improve the problem-solving ability of class VIII students of MTs Hikmatusysyarief NW Salut Narmada.

scholarly journals Penyelesaian Sistem Persamaan Linier Fully Fuzzy Menggunakan Metode Dekomposisi Nilai Singular (SVD)

2018 ◽  
Vol 4 (2) ◽  
pp. 143-149
Author(s):  
Corry Corazon Marzuki ◽  
Agustian` Agustian` ◽  
Dewi Hariati ◽  
Junitis Afmilda ◽  
Nurul Husna ◽  
...  
Keyword(s):  
Singular Value Decomposition ◽  
Linear Equation ◽  
Fuzzy Numbers ◽  
Equation System ◽  
Singular Value ◽  
Linear Equation System ◽  
Single Solution ◽  
Value Decomposition ◽  
Svd Method ◽  
Linear Equation Systems

Linear equation system can be arranged into the AX = B matrix equation. Constants in linear can also contain fuzzy numbers and all their parameters in fuzzy numbers known as fully fuzzy linear equation systems. singular value decomposition (SVD) is a method that decomposes an A matrix into three components of the USVH. The SVD method can be used to find a solution to the fully fuzzy fully linear equation system that is also an inconsistent fully fuzzy linear equation system. The solution obtained from a fully fuzzy linear equation system that is consistent using SVD is a single solution and many solutions. Whereas, the solution obtained from a fully fuzzy linear equation system that is inconsistent using SVD is the best approach solution.

Dynamic interaction of multiple buoyant jets

2012 ◽  
Vol 708 ◽  
pp. 539-575 ◽  
Author(s):  
Adrian C. H. Lai ◽  
Joseph H. W. Lee
Keyword(s):  
Stokes Equations ◽  
Dynamic Pressure ◽  
Unit Length ◽  
Iterative Solution ◽  
Scalar Fields ◽  
Passive Scalar ◽  
Dynamic Interaction ◽  
Ambient Conditions ◽  
Buoyant Jet ◽  
Buoyant Jets

AbstractAn array of closely spaced round buoyant jets interact dynamically due to the pressure field induced by jet entrainment. Mutual jet attraction can result in a significant change in jet trajectories. Jet merging also leads to overlapping of the passive scalar fields associated with the individual jets, resulting in mixing characteristics that are drastically different from those of an independent free jet. A general semi-analytical model for the dynamic interaction of multiple buoyant jets in stagnant ambient conditions is proposed. The external irrotational flow field induced by the buoyant jets is computed by a distribution of point sinks with strength equal to the entrainment per unit length along the unknown jet trajectories and accounting for boundary effects. The buoyant jet trajectories are then determined by an iterative solution of an integral buoyant jet model by tracking the changes in the external entrainment flow and dynamic pressure fields. The velocity and concentration fields of the jet group are obtained by momentum or kinetic energy superposition for merged jets and plumes, respectively. The modelling approach is supported by numerical solution of the Reynolds-averaged Navier–Stokes equations. The model shows that jet merging and mixing can be significantly affected by jet interactions. Model predictions of the multiple jet trajectories, merging height, as well as the centreline velocity and concentration of the buoyant jet group are in good agreement with experimental data for: (i) a clustered momentum jet group; (ii) a turbulent plume pair; and (iii) a rosette buoyant jet group. Dynamic interactions between a jet group are shown to decrease with the addition of an ambient cross-flow.

A finite element method for the Navier-Stokes equations in moving domain with application to hemodynamics of the left ventricle

2017 ◽  
Vol 32 (4) ◽  
Author(s):  
Alexander Danilov ◽  
Alexander Lozovskiy ◽  
Maxim Olshanskii ◽  
Yuri Vassilevski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Left Ventricle ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Computed Tomography Images ◽  
Time Dependent Domain ◽  
Element Method

AbstractThe paper introduces a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method is based on a quasi-Lagrangian formulation of the problem and handling the geometry in a time-explicit way. We prove that numerical solution satisfies a discrete analogue of the fundamental energy estimate. This stability estimate does not require a CFL time-step restriction. The method is further applied to simulation of a flow in a model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

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