DEVELOPMENT OF A HIGHER-ORDER ACCURATE RECONSTRUCTION SCHEME WITH REDUCED LEAST SQUARE MATRIX FOR CONVECTIVE AND DIFFUSIVE FLUX EVALUATION

2009 ◽  
Vol 06 (03) ◽  
pp. 425-446 ◽  
Author(s):  
PRAVEEN NAIR ◽  
T. JAYACHANDRAN ◽  
BHAL CHANDRA PURANIK ◽  
V. UPENDRA BHANDARKAR
Keyword(s):  
Finite Volume ◽  
Higher Order ◽  
Least Square ◽  
Test Facility ◽  
Adaptive Grids ◽  
Computationally Efficient ◽  
Time Step ◽  
Reconstruction Scheme ◽  
The Matrix ◽  
Square Matrix

The development of a higher-order reconstruction scheme with reduced least square matrix is presented. The matrix used in conventional least square based reconstruction schemes for finite volume solvers contains bigger terms. For solution dependent schemes, this matrix has to be inverted for each time step, which is computationally costlier. To overcome this, certain mathematical principles applicable to finite volume formulation, have been used to eliminate a good number of terms appearing in the matrix. In addition, accurate and computationally efficient derivative plug-ins are incorporated to make the formulation generalized so that one can extend it to any order of accuracy. The presence of higher derivative terms in this scheme ensures uniformly higher-order accuracy throughout the flow domain. The reduced matrix can be used for data independent as well as solution dependent reconstruction schemes. Computationally efficient stencil searching algorithm, satisfying physical and topological requirements and capable of handling structured, unstructured, and adaptive grids has been coupled with the scheme. The scheme has been successfully used to simulate flow over blunt cone-flare, NASA B2 nozzle, and high altitude test facility. The solver has shown around 30% saving in least square matrix evaluation time.

A Hybrid Finite Volume/PDF Monte Carlo Method to Capture Sharp Gradients in Unstructured Grids

2000 ◽  
Author(s):  
Genong Li ◽  
Michael F. Modest
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Finite Volume ◽  
Solution Procedure ◽  
Reactive Flow ◽  
Diffusion Combustion ◽  
Monte Carlo Code ◽  
Adaptive Grids ◽  
Time Step ◽  
Particle Tracing

Abstract The hybrid finite volume/PDF Monte Carlo method has both the advantages of the finite volume method’s efficiency in solving flow fields and the PDF method’s exactness in dealing with chemical reactions. It is, therefore, increasingly used in turbulent reactive flow calculations. In order to resolve the sharp gradients of flow velocities and/or scalars, fine grids or unstructured solution -adaptive grids have to be used in the finite volume code. As a result, the calculation domain is covered by a grid system with very large variations in cell size. Such grids present a challenge for a combined PDF/Monte Carlo code. To date, PDF calculations have generally been carried out with large cells, which assure that each cell has a statistically meaningful number of particles. Smaller cells would lead to smaller numbers of particles and correspondingly larger statistical errors. In this paper, a particle tracing scheme with adaptive time step and particle splitting and combination is developed, which allows the PDF/Monte Carlo code to use any grid that is constructed in the finite volume code. This relaxation of restrictions on the grid makes it possible to couple PDF/Monte Carlo methods to all popular commercial CFD codes and, consequently, extend existing CFD codes’ capability to simulate turbulent reactive flow in a more accurate way. To illustrate the solution procedure, a PDF/ Monte Carlo code is combined with FLUENT to solve a turbulent diffusion combustion problem in an axisymmetric channel.

scholarly journals A stable finite-volume method for scalar field dark matter

2019 ◽  
Vol 489 (2) ◽  
pp. 2367-2376 ◽  
Author(s):  
Philip F Hopkins
Keyword(s):  
Dark Matter ◽  
Scalar Field ◽  
Finite Volume ◽  
Higher Order ◽  
Test Problems ◽  
Computationally Efficient ◽  
Mesh Free ◽  
Light Scalar ◽  
Gravitating Systems ◽  
Volume Method

ABSTRACT We describe and test a family of new numerical methods to solve the Schrödinger equation in self-gravitating systems, e.g. Bose–Einstein condensates or ‘fuzzy’/ultra-light scalar field dark matter. The methods are finite-volume Godunov schemes with stable, higher order accurate gradient estimation, based on a generalization of recent mesh-free finite-mass Godunov methods. They couple easily to particle-based N-body gravity solvers (with or without other fluids, e.g. baryons), are numerically stable, and computationally efficient. Different sub-methods allow for manifest conservation of mass, momentum, and energy. We consider a variety of test problems and demonstrate that these can accurately recover solutions and remain stable even in noisy, poorly resolved systems, with dramatically reduced noise compared to some other proposed implementations (though certain types of discontinuities remain challenging). This is non-trivial because the ‘quantum pressure’ is neither isotropic nor positive definite and depends on higher order gradients of the density field. We implement and test the method in the code gizmo.

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.

A Useful Curve Fitting Method for Concrete Uniaxial Compressive Experiment Data

2011 ◽  
Vol 291-294 ◽  
pp. 1015-1020 ◽  
Author(s):  
Chong Jin ◽  
Hong Wang ◽  
Xiao Zhou Xia
Keyword(s):  
Experimental Data ◽  
Orthogonal Polynomial ◽  
Least Square ◽  
Base Function ◽  
Concrete Material ◽  
Fitting Method ◽  
Orthogonal Base ◽  
The Matrix ◽  
Uniform Function ◽  
Curve Fitting Method

Based on the superiority avoiding the matrix equation to be morbid for those fitting functions constructed by orthogonal base, the Legendre orthogonal polynomial is adopted to fit the experimental data of concrete uniaxial compression stress-strain curves under the frame of least-square. With the help of FORTRAN programming, 3 series of experimental data is fitted. And the fitting effect is very satisfactory when the item number of orthogonal base is not less than 5. What’s more, compared with those piecewise fitting functions, the Legendre orthogonal polynomial fitting function obtained can be introduced into the nonlinear harden-soften character of concrete constitute law more convenient because of its uniform function form and continuous derived feature. And the fitting idea by orthogonal base function will provide a widely road for studying the constitute law of concrete material.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

scholarly journals A Computationally Efficient Shallow Water Model for Mixed Cohesive and Non-Cohesive Sediment Transport in the Yangtze Estuary

Water ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1435
Author(s):  
Peng Hu ◽  
Junyu Tao ◽  
Aofei Ji ◽  
Wei Li ◽  
Zhiguo He
Keyword(s):  
Sediment Transport ◽  
Yangtze Estuary ◽  
Water Model ◽  
Cohesive Sediments ◽  
Shallow Water Model ◽  
Computationally Efficient ◽  
Time Step ◽  
The Yangtze Estuary ◽  
Erosion Coefficient ◽  
Cohesive Sediment Transport

In this paper, a computationally efficient shallow water model is developed for sediment transport in the Yangtze estuary by considering mixed cohesive and non-cohesive sediment transport. It is firstly shown that the model is capable of reproducing tidal-hydrodynamics in the estuarine region. Secondly, it is demonstrated that the observed temporal variation of suspended sediment concentration (SSC) for mixed cohesive and non-cohesive sediments can be well-captured by the model with calibrated parameters (i.e., critical shear stresses for erosion/deposition, erosion coefficient). Numerical comparative studies indicate that: (1) consideration of multiple sediment fraction (both cohesive and non-cohesive sediments) is important for accurate modeling of SSC in the Yangtze Estuary; (2) the critical shear stress and the erosion coefficient is shown to be site-dependent, for which intensive calibration may be required; and (3) the Deepwater Navigation Channel (DNC) project may lead to enhanced current velocity and thus reduced sediment deposition in the North Passage of the Yangtze Estuary. Finally, the implementation of the hybrid local time step/global maximum time step (LTS/GMaTS) (using LTS to update the hydro-sediment module but using GMaTS to update the morphodynamic module) can lead to a reduction of as high as 90% in the computational cost for the Yangtze Estuary. This advantage, along with its well-demonstrated quantitative accuracy, indicates that the present model should find wide applications in estuarine regions.

scholarly journals Eigenbranes in Jackiw-Teitelboim gravity

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Andreas Blommaert ◽  
Thomas G. Mertens ◽  
Henri Verschelde
Keyword(s):  
Finite Volume ◽  
Chaotic System ◽  
Path Integral ◽  
Hermitian Matrices ◽  
The Matrix ◽  
Integral Contour

Abstract It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

scholarly journals Coupled THM and Matrix Stability Modeling of Hydroshearing Stimulation in a Coupled Fracture-Matrix Hot Volcanic System

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Xiao ◽  
Jianchun Guo ◽  
Hehua Wang ◽  
Lize Lu ◽  
John McLennan ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Thermal Conduction ◽  
Dominant Role ◽  
Injection Well ◽  
Time Step ◽  
Volcanic System ◽  
Induced Stress ◽  
The Matrix ◽  
Matrix Stability ◽  
The Stability

A coupled thermal-hydraulic-mechanical (THM) model is developed to simulate the combined effect of fracture fluid flow, heat transfer from the matrix to injected fluid, and shearing dilation behaviors in a coupled fracture-matrix hot volcanic reservoir system. Fluid flows in the fracture are calculated based on the cubic law. Heat transfer within the fracture involved is thermal conduction, thermal advection, and thermal dispersion; within the reservoir matrix, thermal conduction is the only mode of heat transfer. In view of the expansion of the fracture network, deformation and thermal-induced stress model are added to the matrix node’s in situ stress environment in each time step to analyze the stability of the matrix. A series of results from the coupled THM model, induced stress, and matrix stability indicate that thermal-induced aperture plays a dominant role near the injection well to enhance the conductivity of the fracture. Away from the injection well, the conductivity of the fracture is contributed by shear dilation. The induced stress has the maximum value at the injection point; the deformation-induced stress has large value with smaller affected range; on the contrary, thermal-induced stress has small value with larger affected range. Matrix stability simulation results indicate that the stability of the matrix nodes may be destroyed; this mechanism is helpful to create complex fracture networks.

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

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