scholarly journals Accuracy analysis and improvement for the cell-centered scheme with the quasi-one-dimensional reconstriction

2021 ◽  
pp. 1-32 ◽  
Author(s):  
Pavel Alexeevisch Bakhvalov
Keyword(s):  
Transport Equation ◽  
Finite Volume ◽  
Constant Velocity ◽  
Truncation Error ◽  
Finite Volume Scheme ◽  
Accuracy Analysis ◽  
Third Order ◽  
One Dimensional ◽  
The Third ◽  
Dimensional Reconstruction

We study the cell-centered finite-volume scheme with the quasi-one-dimensional reconstruction. For the model transport equation with a constant velocity, we prove that on translationally-invariant (TI) triangular meshes it possesses the second order of the truncation error and, if the solution is steady, the third order of the solution error. We offer the modification possessing the third order of the solution error on TI-meshes for unsteady solutions also and verify its accuracy on unstructured meshes.

scholarly journals On the use of a cell-centered finite-volume scheme on prismatic meshes in boundary layers

2021 ◽  
pp. 1-44
Author(s):  
Pavel Alexeevisch Bakhvalov
Keyword(s):  
Reynolds Number ◽  
Finite Volume ◽  
High Reynolds Number ◽  
Finite Volume Scheme ◽  
Wall Surface ◽  
One Dimensional ◽  
Volume Scheme ◽  
Dimensional Reconstruction ◽  
A Cell ◽  
High Reynolds

We consider the cell-centered finite-volume scheme with the quasi-one-dimensional reconstruction and generalize it to anisotropic prismatic meshes suitable for high-Reynolds-number problems. We offer a new algorithm of flux computation based on the reconstruction along the wall surface, whereas in the original schemes it was along the tangent to the wall surface. We also study how does the curvature of mesh elements influence the accuracy if taken into account.

Optimal Third-Order Symplectic Integration Modeling of Seismic Acoustic Wave Propagation

2020 ◽  
Vol 110 (2) ◽  
pp. 754-762 ◽  
Author(s):  
Chuan Li ◽  
Jianxin Liu ◽  
Bo Chen ◽  
Ya Sun
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Seismic Wave ◽  
Truncation Error ◽  
Seismic Wave Propagation ◽  
Symplectic Integrator ◽  
Third Order ◽  
The Third ◽  
2D And 3D

ABSTRACT Seismic wavefield modeling based on the wave equation is widely used in understanding and predicting the dynamic and kinematic characteristics of seismic wave propagation through media. This article presents an optimal numerical solution for the seismic acoustic wave equation in a Hamiltonian system based on the third-order symplectic integrator method. The least absolute truncation error analysis method is used to determine the optimal coefficients. The analysis of the third-order symplectic integrator shows that the proposed scheme exhibits high stability and minimal truncation error. To illustrate the accuracy of the algorithm, we compare the numerical solutions generated by the proposed method with the theoretical analysis solution for 2D and 3D seismic wave propagation tests. The results show that the proposed method reduced the phase error to the eighth-order magnitude accuracy relative to the exact solution. These simulations also demonstrated that the proposed third-order symplectic method can minimize numerical dispersion and preserve the waveforms during the simulation. In addition, comparing different central frequencies of the source and grid spaces (90, 60, and 20 m) for simulation of seismic wave propagation in 2D and 3D models using symplectic and nearly analytic discretization methods, we deduce that the suitable grid spaces are roughly equivalent to between one-fourth and one-fifth of the wavelength, which can provide a good compromise between accuracy and computational cost.

THE THIRD ORDER NONLINEAR SUSCEPTIBILITY OF INTERACTING ONE-DIMENSIONAL FRENKEL EXCITONS

2001 ◽  
Vol 15 (28n30) ◽  
pp. 3809-3812 ◽  
Author(s):  
H. Ishihara ◽  
T. Amakata
Keyword(s):  
Exciton State ◽  
Nonlinear Susceptibility ◽  
State Transitions ◽  
Third Order ◽  
One Dimensional ◽  
Long Wavelength ◽  
The Third ◽  
Wavelength Approximation ◽  
Frenkel Excitons ◽  
Third Order Nonlinear Susceptibility

An analytical expression of the third order nonlinear susceptibility χ(3) has been derived rigorously for a system of interacting Frenkel excitons in a one-dimensional chain of size N with the periodic boundary conditions. It has been clarified that the magnitude of interacting potential between excitons strongly influences the size dependence of χ(3) in the long wavelength approximation, which is well explained in terms of the cancellation effect between the contributions from [ground state] - [one-exciton] transitions and those from [one-exciton] - [two-exciton state] transitions.

scholarly journals A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model

2021 ◽  
Author(s):  
Graham Baird ◽  
Endre Suli
Keyword(s):  
Finite Volume ◽  
Numerical Scheme ◽  
Truncation Error ◽  
Approximate Solutions ◽  
Fragmentation Model ◽  
Finite Volume Scheme ◽  
Finite Volume Schemes ◽  
Operator Semigroups ◽  
Volume Scheme ◽  
Fragmentation Equation

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford-Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate L1 space to a weak solution to the problem. By applying the methods and theory of operator semigroups, we are able to show that these weak solutions are unique and necessarily classical (differentiable) solutions, a degree of regularity not generally established when finite volume schemes are applied to such problems. Furthermore, this approach enabled us to derive a bound for the error induced by the truncation of the mass domain, and also establish the convergence of the truncated solutions as the truncation point is increased without bound. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence as the mesh is refined, whilst also verifying the bound on the truncation error.

Modulated one-dimensional structure of [Cd(NH3)3Ni(CN)4]

2005 ◽  
Vol 61 (3) ◽  
pp. 280-286 ◽  
Author(s):  
Václav Petříček ◽  
Michal Dušek ◽  
Juraj Černák
Keyword(s):  
Crystal Structure ◽  
Data Collection ◽  
Dimensional Structure ◽  
Third Order ◽  
One Dimensional ◽  
The Third ◽  
Incommensurate Structure ◽  
Point Detector

The crystal structure of [Cd(NH3)3Ni(CN)4] has been solved and refined as an incommensurate structure in four-dimensional superspace. The Xcalibur point-detector diffractometer was used for data collection up to the third-order satellites. The structure was refined from 3496 observed reflections to a final R value of 0.0371. The modulation strongly affects all the atoms of the structure. The dominating feature of the modulation is the incommensurate alternation between octahedrally and tetrahedrally coordinated Cd atoms in the electroneutral chains [—Cd(NH3) n —NC—Ni(CN)2—CN—]∞ (n alternates between 2 and 4). The chain is propagated along the a axis, in which the [Cd(NH3) n ]2+ cations are linked by square [Ni(CN)4]2− anions. The atomic positional modulations, having mainly a transversal character in the c-direction, exhibit a switching character that can be described by the crenel and sawtooth functions. The effectively pentagonal coordinated cadmium positions were detected in the intermediate regions.

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