scholarly journals On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2184
Author(s):  
Alexander Zlotnik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Gas Dynamics ◽  
System Of Equations ◽  
Time Step ◽  
Spatial Direction ◽  
Barotropic Gas ◽  
Rectangular Mesh ◽  
2D And 3D

We deal with 2D and 3D barotropic gas dynamics system of equations with two viscous regularizations: so-called quasi-gas dynamics (QGD) and quasi-hydrodynamics (QHD) ones. The system is linearized on a constant solution with any velocity, and an explicit two-level in time and symmetric three-point in each spatial direction finite-difference scheme on the uniform rectangular mesh is considered for the linearized system. We study L2-dissipativity of solutions to the Cauchy problem for this scheme by the spectral method and present a criterion in the form of a matrix inequality containing symbols of symmetric matrices of convective and regularizing terms. Analyzing these inequality and matrices, we also derive explicit sufficient conditions and necessary conditions in the Courant-type form which are rather close to each other. For the QHD regularization, such conditions are derived for the first time in 2D and 3D cases, whereas, for the QGD regularization, they improve those that have recently been obtained. Explicit formulas for a scheme parameter that guarantee taking the maximal time step are given for these conditions. An important moment is a new choice of an “average” spatial mesh step ensuring the independence of the conditions from the ratios of the spatial mesh steps and, for the QGD regularization, from the Mach number as well.

scholarly journals FINITE-DIFFERENCE ANALYSIS FOR THE LINEAR THERMOPOROELASTICITY PROBLEM AND ITS NUMERICAL RESOLUTION BY MULTIGRID METHODS

2012 ◽  
Vol 17 (2) ◽  
pp. 227-244 ◽  
Author(s):  
Natalia Boal ◽  
Francisco Jos´e Gaspar ◽  
Francisco Lisbona ◽  
Petr Vabishchevich
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
System Of Equations ◽  
Convergence Properties ◽  
Discrete Energy ◽  
Numerical Resolution ◽  
Multigrid Convergence

This paper deals with the numerical solution of a two-dimensional thermoporoelasticity problem using a finite-difference scheme. Two issues are discussed: stability and convergence in discrete energy norms of the finite-difference scheme are proved, and secondly, a distributive smoother is examined in order to find a robust and efficient multigrid solver for the corresponding system of equations. Numerical experiments confirm the convergence properties of the proposed scheme, as well as fast multigrid convergence.

scholarly journals Application of Adjoint Data Assimilation Method to Atmospheric Aerosol Transport Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Minjie Xu ◽  
Kai Fu ◽  
Xianqing Lv
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Large Scale ◽  
Large Time ◽  
Computational Time ◽  
Aerosol Transport ◽  
Time Step ◽  
Adjoint Assimilation ◽  
Transport Problems

We propose combining the adjoint assimilation method with characteristic finite difference scheme (CFD) to solve the aerosol transport problems, which can predict the distribution of atmospheric aerosols efficiently by using large time steps. Firstly, the characteristic finite difference scheme (CFD) is tested to compute the Gaussian hump using large time step sizes and is compared with the first-order upwind scheme (US1) using small time steps; the US1 method gets E2 error of 0.2887 using Δt=1/450, while CFD method gets a much smaller E2 of 0.2280 using a much larger time step Δt=1/45. Then, the initial distribution of PM2.5 concentration is inverted by the adjoint assimilation method with CFD and US1. The adjoint assimilation method with CFD gets better accuracy than adjoint assimilation method with US1 while adjoint assimilation method with CFD costs much less computational time. Further, a real case of PM2.5 concentration distribution in China during the APEC 2014 is simulated by using adjoint assimilation method with CFD. The simulation results are in good agreement with the observed values. The adjoint assimilation method with CFD can solve large scale aerosol transport problem efficiently.

A NEW LINEAR AND CONSERVATIVE FINITE DIFFERENCE SCHEME FOR THE GROSS–PITAEVSKII EQUATION WITH ANGULAR MOMENTUM ROTATION

2019 ◽  
Vol 61 (02) ◽  
pp. 204-232
Author(s):  
JIN CUI ◽  
WENJUN CAI ◽  
CHAOLONG JIANG ◽  
YUSHUN WANG
Keyword(s):  
Angular Momentum ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Mesh Size ◽  
Pitaevskii Equation ◽  
Time Step ◽  
Lifting Technique ◽  
Conservative Finite Difference Scheme

A new linear and conservative finite difference scheme which preserves discrete mass and energy is developed for the two-dimensional Gross–Pitaevskii equation with angular momentum rotation. In addition to the energy estimate method and mathematical induction, we use the lifting technique as well as some well-known inequalities to establish the optimal $H^{1}$ -error estimate for the proposed scheme with no restrictions on the grid ratio. Unlike the existing numerical solutions which are of second-order accuracy at the most, the convergence rate of the numerical solution is proved to be of order $O(h^{4}+\unicode[STIX]{x1D70F}^{2})$ with time step $\unicode[STIX]{x1D70F}$ and mesh size $h$ . Numerical experiments have been carried out to show the efficiency and accuracy of our new method.

On Some Finite Difference Scheme for Gas Dynamics Equations

2018 ◽  
Vol 73 (4) ◽  
pp. 143-149
Author(s):  
A. V. Zvyagin ◽  
G. M. Kobelkov ◽  
M. A. Lozhnikov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Gas Dynamics ◽  
Gas Dynamics Equations
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