scholarly journals Two-Dimensional-One-Dimensional Alternating Direction Schemes for Coastal Systems Convection-Diffusion Problems

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3267
Author(s):  
Alexander Sukhinov ◽  
Valentina Sidoryakina
Keyword(s):  
Information Exchange ◽  
Problem Solution ◽  
Two Dimensional ◽  
Splitting Scheme ◽  
Time Step ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Coastal Systems ◽  
Alternating Direction ◽  
Matter Transport

The initial boundary value problem for the 3D convection-diffusion equation corresponding to the mathematical model of suspended matter transport in coastal marine systems and extended shallow water bodies is considered. Convective and diffusive transport operators in horizontal and vertical directions for this type of problem have significantly different physical and spectral properties. In connection with the above, a two-dimensional–one-dimensional splitting scheme has been built—a three-dimensional analog of the Peaceman–Rachford alternating direction scheme, which is suitable for the operational suspension spread prediction in coastal systems. The paper has proved the theorem of stability solution with respect to the initial data and functions of the right side, in the case of time-independent operators in special energy norms determined by one of the splitting scheme operators. The accuracy has been investigated, which, as in the case of the Peaceman–Rachford scheme, with the special definition of boundary conditions on a fractional time step, is the value of the second order in dependency of time and spatial steps. The use of this approach makes it possible to obtain parallel algorithms for solving grid convection-diffusion equations which are economical in the sense of total time of problem solution on multiprocessor systems, which includes time for arithmetic operations realization and the one required to carry of information exchange between processors.

Enhanced Explicit Scheme to Solve Transient Heat Conduction Problem

2007 ◽  
Vol 2 (1) ◽  
Author(s):  
Ganesh Hegde ◽  
Madhu Gattumane
Keyword(s):  
Heat Conduction ◽  
Correction Factor ◽  
Truncation Error ◽  
Transient Heat Conduction ◽  
Explicit Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Time Step ◽  
One Dimensional ◽  
Partial Derivatives

Improvement in accuracy without sacrificing stability and convergence of the solution to unsteady diffusion heat transfer problems by computational method of enhanced explicit scheme (EES), has been achieved and demonstrated, through transient one dimensional and two dimensional heat conduction. The truncation error induced in the explicit scheme using finite difference technique is eliminated by optimization of partial derivatives in the Taylor series expansion, by application of interface theory developed by the authors. This theory, in its simple terms gives the optimum values to the decision vectors in a redundant linear equation. The time derivatives and the spatial partial derivatives in the transient heat conduction, take the values depending on the time step chosen and grid size assumed. The time correction factor and the space correction factor defined by step sizes govern the accuracy, stability and convergence of EES. The comparison of the results of EES with analytical results, show decreased error as compared to the result of explicit scheme. The paper has an objective of reducing error in the explicit scheme by elimination of truncation error introduced by neglecting the higher order terms in the expansion of the governing function. As the pilot examples of the exercise, the implementation is aimed at solving one-dimensional and two-dimensional problems of transient heat conduction and compared with the results cited in the referred literature.

scholarly journals A two-dimensional problem of filtration theory

1985 ◽  
Vol 7 (1) ◽  
pp. 14-19
Author(s):  
Dao Minh Ngoc
Keyword(s):  
Free Surface ◽  
Ground Water ◽  
Water Flow ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Filtration Theory ◽  
Ground Water Flow ◽  
One Dimensional ◽  
Convection Diffusion

The problem about free surface of ground Water flow is scaled Completely with considering the convection diffusion of soluble matter and the matter exchange with ground. A local one-dimensional method of changeable directions is used to perform calculations.

Calibration of a 1D/1D urban flood model using 1D/2D model results in the absence of field data

2011 ◽  
Vol 64 (5) ◽  
pp. 1016-1024 ◽  
Author(s):  
J. Leandro ◽  
S. Djordjević ◽  
A. S. Chen ◽  
D. A. Savić ◽  
M. Stanić
Keyword(s):  
Field Data ◽  
Computational Time ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Time Step ◽  
Urban Flood ◽  
One Dimensional ◽  
1D Model ◽  
Surface Network ◽  
2D Model

Recently increased flood events have been prompting researchers to improve existing coupled flood-models such as one-dimensional (1D)/1D and 1D/two-dimensional (2D) models. While 1D/1D models simulate sewer and surface networks using a one-dimensional approach, 1D/2D models represent the surface network by a two-dimensional surface grid. However their application raises two issues to urban flood modellers: (1) stormwater systems planning/emergency or risk analysis demands for fast models, and the 1D/2D computational time is prohibitive, (2) and the recognized lack of field data (e.g. Hunter et al. (2008)) causes difficulties for the calibration/validation of 1D/1D models. In this paper we propose to overcome these issues by calibrating a 1D/1D model with the results of a 1D/2D model. The flood-inundation results show that: (1) 1D/2D results can be used to calibrate faster 1D/1D models, (2) the 1D/1D model is able to map the 1D/2D flood maximum extent well, and the flooding limits satisfactorily in each time-step, (3) the 1D/1D model major differences are the instantaneous flow propagation and overestimation of the flood-depths within surface-ponds, (4) the agreement in the volume surcharged by both models is a necessary condition for the 1D surface-network validation and (5) the agreement of the manholes discharge shapes measures the fitness of the calibrated 1D surface-network.

scholarly journals Local Two-Dimensional Splitting Schemes for 3D Suspended Matter Transport Problem Parallel Solution

2021 ◽  
pp. 38-53
Author(s):  
Alexander Sukhinov ◽  
◽  
Alexander Chistyakov ◽  
Valentina Sidoryakina ◽  
Sofya Protsenko ◽  
...  
Keyword(s):  
Suspended Matter ◽  
Vertical Direction ◽  
Two Dimensional ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Matter Transport ◽  
Dimensional Splitting ◽  
Convection Problem ◽  
The One ◽  
Diffusion Convection

A 3D model of suspended matter transport in coastal marine systems is considered, which takes into account many factors, including the hydraulic size or the rate of particle deposition, the propagation of suspended matter, sedimentation, the intensity of distribution of suspended matter sources, etc. The difference operators of diffusion transport in the horizontal and vertical directions for this problem have significantly different characteristic spatiotemporal scales of processes, as well as spectra. With typical sampling, applied to shallow-water systems in the South of Russia (the Sea of Azov, the Tsimlyansk reservoir), the steps in horizontal directions are 200-1000 meters, the coefficients of turbulent exchange (turbulent diffusion) are (103-104) m2/sec; in the vertical direction - - - steps of 0.1 m-1 m, and the coefficients of microturbulent exchange in the vertical — (0.1-1) m2/sec. If we focus on the use of explicit locally twodimensional - - - locally one-dimensional splitting schemes, then the permissible values of the time step for a two-dimensional problem will be about 10-100 seconds, and for a one-dimensional problem in the vertical direction - - - 0.1 – 1 sec. This motivates us to construct an additive locally-two-dimensional-locallyonedimensional splitting scheme in geometric directions. The paper describes a parallel algorithm that uses both explicit and implicit schemes to approximate the two-dimensional diffusion-convection problem in horizontal directions and the one-dimensional diffusion-convection problem in the vertical direction. The two-dimensional implicit diffusion-convection problem in horizontal directions is numerically solved by the adaptive alternating-triangular method. The numerical implementation of the one-dimensional diffusion-convection problem in the vertical direction is carried out by a sequential run-through method for a series of independent one-dimensional three-point problems in the vertical direction on a given layer. To increase the efficiency of parallel calculations, the decomposition of the calculated spatial grid and all grid data in one or two spatial directions - in horizontal directions-is also performed. The obtained algorithms are compared taking into account the permissible values of time steps and the actual time spent on performing calculations and exchanging information on each time layer.

scholarly journals MagIC v5.10: a two-dimensional MPI distribution for pseudo-spectral magneto hydrodynamics simulations in spherical geometry

2021 ◽  
Author(s):  
Rafael Lago ◽  
Thomas Gastine ◽  
Tilman Dannert ◽  
Markus Rampp ◽  
Johannes Wicht
Keyword(s):  
High Performance ◽  
Spherical Geometry ◽  
Distribution Data ◽  
Two Dimensional ◽  
Data Layout ◽  
Time Step ◽  
One Dimensional ◽  
Dimensional Distribution ◽  
Grid Points ◽  
Pseudo Spectral

Abstract. We discuss two parallelization schemes for MagIC, an open-source, high-performance, pseudo-spectral code for the numerical solution of the magneto hydrodynamics equations in a rotating spherical shell. MagIC calculates the non-linear terms on a numerical grid in spherical coordinates while the time step updates are performed on radial grid points with a spherical harmonic representation of the lateral directions. Several transforms are required to switch between the different representations. The established hybrid implementation of MagIC uses MPI-parallelization in radius and relies on existing fast spherical transforms using OpenMP. Our new two-dimensional MPI decomposition implementation also distributes the latitudes or the azimuthal wavenumbers across the available MPI tasks/compute cores. We discuss several non-trivial algorithmic optimizations and the different data distribution layouts employed by our scheme. In particular, the two-dimensional distribution data layout yields a code that strongly scales well beyond the limit of the current one-dimensional distribution. We also show that the two-dimensional distribution implementation, although not yet fully optimized, can already be faster than the existing finely optimized hybrid implementation when using many thousands of CPU cores. Our analysis indicates that the two-dimensional distribution variant can be further optimized to also surpass the performance of the one-dimensional distribution for a few thousand cores.

scholarly journals MagIC v5.10: a two-dimensional message-passing interface (MPI) distribution for pseudo-spectral magnetohydrodynamics simulations in spherical geometry

2021 ◽  
Vol 14 (12) ◽  
pp. 7477-7495
Author(s):  
Rafael Lago ◽  
Thomas Gastine ◽  
Tilman Dannert ◽  
Markus Rampp ◽  
Johannes Wicht
Keyword(s):  
Message Passing ◽  
Message Passing Interface ◽  
Distribution Data ◽  
Two Dimensional ◽  
Data Layout ◽  
Time Step ◽  
One Dimensional ◽  
Hybrid Parallelization ◽  
Dimensional Distribution ◽  
Pseudo Spectral

Abstract. We discuss two parallelization schemes for MagIC, an open-source, high-performance, pseudo-spectral code for the numerical solution of the magnetohydrodynamics equations in a rotating spherical shell. MagIC calculates the non-linear terms on a numerical grid in spherical coordinates, while the time step updates are performed on radial grid points with a spherical harmonic representation of the lateral directions. Several transforms are required to switch between the different representations. The established hybrid parallelization of MagIC uses message-passing interface (MPI) distribution in radius and relies on existing fast spherical transforms using OpenMP. Our new two-dimensional MPI decomposition implementation also distributes the latitudes or the azimuthal wavenumbers across the available MPI tasks and compute cores. We discuss several non-trivial algorithmic optimizations and the different data distribution layouts employed by our scheme. In particular, the two-dimensional distribution data layout yields a code that strongly scales well beyond the limit of the current one-dimensional distribution. We also show that the two-dimensional distribution implementation, although not yet fully optimized, can already be faster than the existing finely optimized hybrid parallelization when using many thousands of CPU cores. Our analysis indicates that the two-dimensional distribution variant can be further optimized to also surpass the performance of the one-dimensional distribution for a few thousand cores.

Single Grid Error Estimation Using Error Transport Equation

2004 ◽  
Vol 126 (5) ◽  
pp. 778-790 ◽  
Author(s):  
Ismail Celik ◽  
Gusheng Hu
Keyword(s):  
Diffusion Equation ◽  
Transport Equation ◽  
Mesh Size ◽  
Taylor Series Expansion ◽  
Discretization Error ◽  
Two Dimensional ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Error Transport Equation

This paper presents an approach to quantify the discretization error as well as other errors related to mesh size using the error transport equation (ETE) technique on a single grid computation. The goal is to develop a generalized algorithm that can be used in conjunction with computational fluid dynamics (CFD) codes to quantify the discretization error in a selected process variable. The focus is on applications where the conservation equations are solved for primitive variables, such as velocity, temperature and concentration, using finite difference and/or finite volume methods. An error transport equation (ETE) is formulated. A generalized source term for the ETE is proposed based on the Taylor series expansion and accessible influence coefficients in the discretized equation. Representative examples, i.e., one-dimensional convection diffusion equation, two-dimensional Poisson equation, two-dimensional convection diffusion equation, and non-linear one-dimensional Burger’s equation are presented to verify this method and elucidate its properties. Discussions are provided to address the significance and possible potential applications of this method to Navier-Stokes solvers.

A Hybrid Variational Multiscale Element-Free Galerkin Method for Convection-Diffusion Problems

2019 ◽  
Vol 11 (07) ◽  
pp. 1950063 ◽  
Author(s):  
Jufeng Wang ◽  
Fengxin Sun
Keyword(s):  
Splitting Method ◽  
Two Dimensional ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Variational Multiscale ◽  
Discrete Equations ◽  
Element Free Galerkin ◽  
Oscillating Solutions ◽  
Element Free ◽  
Diffusion Problems

By coupling the dimension splitting method (DSM) and the variational multiscale element-free Galerkin (VMEFG) method, a hybrid variational multiscale element-free Galerkin (HVMEFG) method is developed for the two-dimensional convection-diffusion problems. In the HVMEFG method, the two-dimensional problem is converted into a battery of one-dimensional problems by the DSM. Combining the non-singular improved interpolating moving least-squares (IIMLS) method, the VMEFG method is used to obtain the discrete equations of the one-dimensional problems on the splitting plane. Then, final discretized equations of the entire convection-diffusion problems are assembled by the IIMLS method. The HVMEFG method has high accuracy and efficiency. Numerical examples show that the HVMEFG method can obtain non-oscillating solutions and has higher efficiency and accuracy than the EFG and VMEFG methods for convection-diffusion problems.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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