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

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.

2019 ◽  
Vol 61 (02) ◽  
pp. 204-232
Author(s):  
JIN CUI ◽  
WENJUN CAI ◽  
CHAOLONG JIANG ◽  
YUSHUN WANG

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.


2017 ◽  
Vol 21 (6 Part A) ◽  
pp. 2327-2336 ◽  
Author(s):  
Ernesto Hernandez ◽  
José Otero ◽  
Rubén Santiago ◽  
Raúl Martinez ◽  
Francisco Castillo ◽  
...  

Over a finite 1-D specimen containing two phases of a pure substance, it has been shown that the liquid-solid interface motion exhibits parabolic behavior at small time intervals. We study the interface behavior over a finite domain with homogeneous Dirichlet boundary conditions for large time intervals, where the interface motion is not parabolic due to finite size effects. Given the physical nature of the boundary conditions, we are able to predict exactly the interface position at large time values. These predictions, which to the best of our knowledge, are not found in the literature, were confirmed by using the heat balance integral method of Goodman and a non-classical finite difference scheme. Using heat transport theory, it is shown as well, that the temperature profile within the specimen is exactly linear and independent of the initial profile in the asymptotic time limit. The physics of heat transport provides a powerful tool that is used to fine tune the numerical methods. We also found that in order to capture the physical behavior of the interface, it was necessary to develop a new non-classical finite difference scheme that approaches asymptotically to the predicted interface position. We offer some numerical examples where the predicted effects are illustrated, and finally we test our predictions with the heat balance integral method and the non-classical finite difference scheme by studying the liquid-solid phase transition in aluminum.


Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. A93-A98 ◽  
Author(s):  
Yingjie Gao ◽  
Jinhai Zhang ◽  
Zhenxing Yao

The explicit finite-difference scheme is popular for solving the wave equation in the field of seismic exploration due to its simplicity in numerical implementation. However, its maximum time step is strictly restricted by the Courant-Friedrichs-Lewy (CFL) stability limit, which leads to a heavy computational burden in the presence of small-scale structures and high-velocity targets. We remove the CFL stability limit of the explicit finite-difference scheme using the eigenvalue perturbation, which allows us to use a much larger time step beyond the CFL stability limit. For a given time step that is within the CFL stability limit, the eigenvalues of the update matrix would be distributed along the unit circle; otherwise, some eigenvalues would be distributed outside of the unit circle, which introduces unstable phenomena. The eigenvalue perturbation can normalize the unstable eigenvalues and guarantee the stability of the update matrix by using an arbitrary time step. The update matrix can be preprocessed before the numerical simulation, thus retaining the computational efficiency well. We further incorporate the forward time-dispersion transform (FTDT) and the inverse time-dispersion transform (ITDT) to reduce the time-dispersion error caused by using an unusually large time step. Our numerical experiments indicate that the combination of the eigenvalue perturbation, the FTDT method, and the ITDT method can simulate highly accurate waveforms when applying a time step beyond the CFL stability limit. The time step can be extended even toward the Nyquist limit. This means that we could save many iteration steps without suffering from time-dispersion error and stability problems.


Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2184
Author(s):  
Alexander Zlotnik

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.


2017 ◽  
Vol 21 (6 Part B) ◽  
pp. 2699-2708 ◽  
Author(s):  
Jose Otero ◽  
Ernesto Hernandez ◽  
Ruben Santiago ◽  
Raul Martinez ◽  
Francisco Castillo ◽  
...  

In this work, we study the liquid-solid interface dynamics for large time intervals on a 1-D sample, with homogeneous Neumann boundary conditions. In this kind of boundary value problem, we are able to make new predictions about the interface position by using conservation of energy. These predictions are confirmed through the heat balance integral method of Goodman and a generalized non-classical finite difference scheme. Since Neumann boundary conditions imply that the specimen is thermally isolated, through well stablished thermodynamics, we show that the interface behavior is not parabolic, and some examples are built with a novel interface dynamics that is not found in the literature. Also, it is shown that, on a Neumann boundary value problem, the position of the interface at thermodynamic equilibrium depends entirely on the initial temperature profile. The prediction of the interface position for large time values makes possible to fine tune the numerical methods, and given that energy conservation demands highly precise solutions, we found that it was necessary to develop a general non-classical finite difference scheme where a non-homogeneous moving mesh is considered. Numerical examples are shown to test these predictions and finally, we study the phase transition on a thermally isolated sample with a liquid and a solid phase in aluminum.


Geophysics ◽  
2013 ◽  
Vol 78 (1) ◽  
pp. T21-T27 ◽  
Author(s):  
Adel Khalil ◽  
Mohamed Hesham ◽  
Mohamed El-Beltagy

We propose to solve the two-way time domain acoustic wave equation in a generalized Riemannian coordinate system via finite-differences. The coordinate system is defined in such a way that one of its independent variables conforms to the primary wavefront, for example, using a ray coordinate system with the traveltime being one of the coordinates. At each finite-difference time-step, the solution domain is limited to a narrow corridor around the primary wavefront, leading to an increase in the computational performance. A new finite-difference scheme is introduced to stabilize the solution and facilitate its implementation. This new scheme is a blend of the simple explicit and the stable implicit schemes. As a proof of concept, the proposed method is compared to the classical explicit finite-difference scheme performed in Cartesian coordinates on two synthetic velocity models with varying complexities. At a reduced cost, the proposed method produces similar results to the classical one; however, some amplitude differences arise due to various implementation issues. The most direct application for the proposed method is the source side of reverse time migration.


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