dispersion error
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Author(s):  
Yuta Kawai ◽  
Hirofumi Tomita

AbstractRecently, large-eddy simulation (LES) has been increasingly employed in meteorological simulations because it is a promising method for turbulent parameterization. However, it is still difficult to affirm that the numerical accuracy required for a dynamical core is fully understood. In this study, we derived two theoretical criteria for the order of accuracy of the advection term in a typical situation of the atmospheric boundary layer, and demonstrate their validity by numerical experiments. In the targeted grid-spacing of O (10 m), we determined the required order of accuracy as follows: Based on the criterion of the numerical diffusion error, the upwind scheme must have at least seventh-order accuracy. The fourth-order central scheme is barely acceptable with fourth-order explicit diffusion, provided that its coefficient is one or two orders of magnitude smaller than the implicit diffusion coefficient of the third-order upwind scheme. Based on the criterion of numerical dispersion error, at minimum, the seventh or eighth order is required. The dispersion error was indirect for the energy spectra, although we expect it may affect the local turbulence mechanism. We also investigated the effects of temporal discretization for compressible models, and found that relatively lower-order time schemes are available up to the O(10 m) grid spacing if the time step is sufficiently small due to sound wave limitations. The importance of the derived criteria is that the required order of accuracy increases as the grid spacing decreases. This suggests that considerable care should be taken regarding the numerical error problem for future high-resolution LES.


2021 ◽  
Author(s):  
J. Ruano ◽  
A. Vidal ◽  
J. Rigola ◽  
F. Trias

Entropy ◽  
2020 ◽  
Vol 22 (12) ◽  
pp. 1376
Author(s):  
Áron Pozsár ◽  
Mátyás Szücs ◽  
Róbert Kovács ◽  
Tamás Fülöp

The recent results attained from a thermodynamically conceived numerical scheme applied on wave propagation in viscoelastic/rheological solids are generalized here, both in the sense that the scheme is extended to four spacetime dimensions and in the aspect of the virtues of a thermodynamical approach. Regarding the scheme, the arrangement of which quantity is represented where in discretized spacetime, including the question of appropriately realizing the boundary conditions, is nontrivial. In parallel, placing the problem in the thermodynamical framework proves to be beneficial in regards to monitoring and controlling numerical artefacts—instability, dissipation error, and dispersion error. This, in addition to the observed preciseness, speed, and resource-friendliness, makes the thermodynamically extended symplectic approach that is presented here advantageous above commercial finite element software solutions.


Author(s):  
Tamás Fülöp

Rocks exhibit beyond-Hookean, delayed and damped elastic, behaviour (creep, relaxation etc.). In many cases, the Poynting–Thomson–Zener (PTZ) rheological model proves to describe these phenomena successfully. A forecast of the PTZ model is that the dynamic elasticity coefficients are larger than the static (slow-limit) counterparts. This prediction has recently been confirmed on a large variety of rock types. Correspondingly, according to the model, the speed of wave propagation depends on frequency, the high-frequency limit being larger than the low-frequency limit. This frequency dependence can have a considerable influence on the evaluation of various wave-based measurement methods of rock mechanics. As experience shows, commercial finite element softwares are not able to properly describe wave propagation, even for the Hooke model and simple specimen geometries, the seminal numerical artefacts being instability, dissipation error and dispersion error, respectively. This has motivated research on developing reliable numerical methods, which amalgamate beneficial properties of symplectic schemes, their thermodynamically consistent generalization (including contact geometry), and spacetime aspects. The present work reports on new results obtained by such a numerical scheme, on wave propagation according to the PTZ model, in one space dimension. The simulation outcomes coincide nicely with the theoretically obtained phase velocity prediction.


Electronics ◽  
2020 ◽  
Vol 9 (10) ◽  
pp. 1575 ◽  
Author(s):  
Zhen Kang ◽  
Ming Huang ◽  
Weilin Li ◽  
Yufeng Wang ◽  
Fang Yang

A modified precise-integration time-domain (PITD) formulation is presented to model the wave propagation in magnetized plasma based on the auxiliary differential equation (ADE). The most prominent advantage of this algorithm is using a time-step size which is larger than the maximum value of the Courant–Friedrich–Levy (CFL) condition to achieve the simulation with a satisfying accuracy. In this formulation, Maxwell’s equations in magnetized plasma are obtained by using the auxiliary variables and equations. Then, the spatial derivative is approximated by the second-order finite-difference method only, and the precise integration (PI) scheme is used to solve the resulting ordinary differential equations (ODEs). The numerical stability and dispersion error of this modified method are discussed in detail in magnetized plasma. The stability analysis validates that the simulated time-step size of this method can be chosen much larger than that of the CFL condition in the finite-difference time-domain (FDTD) simulations. According to the numerical dispersion analysis, the range of the relative error in this method is 10−6 to 5×10−4 when the electromagnetic wave frequency is from 1 GHz to 100 GHz. More particularly, it should be emphasized that the numerical dispersion error is almost invariant under different time-step sizes which is similar to the conventional PITD method in the free space. This means that with the increase of the time-step size, the presented method still has a lower computational error in the simulations. Numerical experiments verify that the presented method is reliable and efficient for the magnetized plasma problems. Compared with the formulations based on the FDTD method, e.g., the ADE-FDTD method and the JE convolution FDTD (JEC-FDTD) method, the modified algorithm in this paper can employ a larger time step and has simpler iterative formulas so as to reduce the execution time. Moreover, it is found that the presented method is more accurate than the methods based on the FDTD scheme, especially in the high frequency range, according to the results of the magnetized plasma slab. In conclusion, the presented method is efficient and accurate for simulating the wave propagation in magnetized plasma.


2020 ◽  
Vol 17 (2(SI)) ◽  
pp. 0689
Author(s):  
Mohammed Salih ◽  
Fudziah Ismail ◽  
Norazak Senu

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.


IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 199016-199024
Author(s):  
Yong-Dan Kong ◽  
Chu-Bin Zhang ◽  
Hong-Yu Zhang ◽  
Qing-Xin Chu

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1383
Author(s):  
Nie ◽  
Gui ◽  
Chen

The symplectic algorithm can maintain the symplectic structure and intrinsic properties of the system, its cumulative error is small and suitable for multi-step calculation. At present, the widely accepted symplectic operators are obtained by solving the Hamilton equation based on artificial definitions and assumptions in advance. There are inevitable dispersion errors. We solve the equation by pure mathematical derivation without any artificial limitations and assumptions. The way to accurately obtain high-precision symplectic operators greatly reduces the dispersion error from the beginning. The numerical solution of the one-dimensional Schrödinger equation for describing the intrinsic problem of nanodevices is used as an application environment to compare the total energy distribution of the particle wave function in the box, thus verifying the properties of the Symplectic Operator based on Pure Mathematical Derivation by comparing with Finite-Difference Time-Domain (FDTD) and the widely accepted symplectic operator.


2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Haitao Wang ◽  
Xiangyang Zeng ◽  
Ye Lei

Wave-based methods for acoustic simulations within enclosures suffer the numerical dispersion and then usually have evident dispersion error for problems with high wave numbers. To improve the upper limit of calculating frequency for 3D problems, a hybrid smoothed finite element method (hybrid SFEM) is proposed in this paper. This method employs the smoothing technique to realize the reduction of the numerical dispersion. By constructing a type of mixed smoothing domain, the traditional node-based and face-based smoothing techniques are mixed in the hybrid SFEM to give a more accurate stiffness matrix, which is widely believed to be the ultimate cause for the numerical dispersion error. The numerical examples demonstrate that the hybrid SFEM has better accuracy than the standard FEM and traditional smoothed FEMs under the condition of the same basic elements. Moreover, the hybrid SFEM also has good performance on the computational efficiency. A convergence experiment shows that it costs less time than other comparison methods to achieve the same computational accuracy.


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