### Table 4.3 Pre-eigenvalues, targets and computed eigenvalues for the convection-diffusion problem.

1999

Cited by 3

### Table 3.4 The one-dimensional diffusion-convection problem. True error, effectivity indices for various values of the preset tolerance TOL.

2003

Cited by 2

### Table 1. Equations of convective radiation hydrodynamics in di erential form. D=Dt denotes the comoving or substantial derivative with respect to time. Refer to Table 6 for a comprehensive list of symbols. Gas dynamics

"... In PAGE 2: ... The remaining inte- gration of the inner mass mr over the density structure is performed by solving Eq. 1 in Table1 , where we employ mr as the independent variable. Further details concerning the equations of hydrody- namics can be found e.... In PAGE 2: ....2. Radiative transfer Time-dependent radiative transfer is considered in the grey approximation by solving the rst two frequency- integrated moment equations of the radiation eld rep- resenting energy conservation (radiation energy equation, Eq. 5 in Table1 ) and momentum conservation (radiation momentum equation, Eq. 6 in Table 1).... In PAGE 2: ... 5 in Table 1) and momentum conservation (radiation momentum equation, Eq. 6 in Table1 ). As independent variables we use the zeroth moment of the radiation eld J and the rst moment of the radiation eld H, which are proportional to the radiative energy density and the radiative ux, respectively (e.... In PAGE 3: ...ecko et al. (1998) and Koll ath et al. (1998). The basic equation (Eq. 8 in Table1 ) of the Kuh- fu model is a conservation law for the turbulent kinetic energy density !, which serves as the independent variable of the turbulent eld. The essential term is the turbulent driving through buoyancy forces (S !).... In PAGE 3: ... The basic quantity entering S ! is the entropy gradient @s=@r which is related to the Schwarzschild criterion by Eq. 12 ( Table1 ). Note that S ! can be evaluated even if the Schwarzschild crite- rion indicates convective stability.... In PAGE 4: ...6. The model equations In Table1 the complete set of model equations for gas dynamics, radiative transfer and turbulent convection in di erential form are compiled. With regard to the numer- ical method of solution using a conservative volume dis- cretization on an adaptive mesh (cf.... In PAGE 4: ... The corresponding total internal energy equation (Eq. 4 in Table1 ) can be derived by adding the respective energy equations of gas, radiation and convection (Eqns. 4,5 and 8 in Table 1).... In PAGE 4: ... Numerically speaking the initial model has to lie within the convergence radius of the implicit solution scheme. Consequently we use the hydrostatic and local limit of the nonlinear equations given in Table1 to compute the structure of the initial model. This limit is obtained by omitting all time derivatives, setting the gas velocity iden- tically to zero (u 0), and neglecting turbulent pressure and overshooting.... In PAGE 10: ...urb. energy eq. Table 4. Hydrostatic and local limit of the convective radiation hydrodynamics equations given in Table1 . See text for details.... ..."

### Table 1 Diffusion equations used in image processing. The general form of the equations is

1999

"... In PAGE 2: ... JPEG has a characteristic blocking-artefact (Figures 1,2,3 and 6). PDEs of Table1 were tested as perceptually adaptive filters for compression artefact reduction in [8]. The adequate perceptually adaptive filter fell out to be the PMC-AD, since it suppresses noise while performing shape enhancement.... ..."

Cited by 1

### Table 2 summarizes the sampling distributions of the three estimators of the diffusion

"... In PAGE 20: ... If, however, the information about intraday volatility that is revealed by the range but not by absolute or squared returns is useful in the estimation of the model, the sampling properties of the range- based quasi-maximum likelihood estimator could well dominate the sampling properties of the exact maximum likelihood estimator for absolute returns. 14 Comparing the third row of each panel in Table2... In PAGE 21: ...bsolute return as volatility proxy. First, the range-based parameter estimates are more accurate. Second, even for the same parameters values, the approximate normality of the log range yields a more efficient volatility extraction. With this in mind, we summarize in the last two panels of Table2 (and in the last column of Figure 2) the sampling distributions of the average extraction error , which is 1 T j T t apos;1 ( ln Ft amp;ln Ft ) an estimator of the expected extraction error , and the average squared extraction E [ ln Ft amp;ln Ft ] error , which is an estimator of the expected squared extraction error 1 T j T t apos;1 ( ln Ft amp;ln... In PAGE 22: ... Now we discuss the results for a smaller sample size of T = 500 observations and a larger sample size of T = 5000 observations. We show the results for T = 500 in Table 3; they are qualitatively identical to those in Table2 . Quantitatively, however, the relative performance of the quasi-maximum likelihood estimator with the log absolute return as volatility proxy, which was already poor with T = 1000 observations, is much worse with T = 500 observations.... In PAGE 22: ... D We present the results for T = 5000 in Table 4. Qualitatively, they are again identical to the results in Table2 ; quantitatively, the comparative performance of the quasi-maximum likelihood estimator with the log absolute return as volatility proxy is improved in some respects,... ..."

### Table 2 summarizes the sampling distributions of the three estimators of the diffusion

"... In PAGE 18: ... If, however, the information about intraday volatility that is revealed by the range but not by absolute or squared returns is useful in the estimation of the model, the sampling properties of the range- based quasi-maximum likelihood estimator could well dominate the sampling properties of the exact maximum likelihood estimator for absolute returns. 14 Comparing the third row of each panel in Table2... In PAGE 19: ...bsolute return as volatility proxy. First, the range-based parameter estimates are more accurate. Second, even for the same parameters values, the approximate normality of the log range yields a more efficient volatility extraction. With this in mind, we summarize in the last two panels of Table2 (and in the last column of Figure 2) the sampling distributions of the average extraction error , which is 1 T j T t apos;1 ( ln Ft amp;ln Ft ) an estimator of the expected extraction error , and the average squared extraction E [ ln Ft amp;ln Ft ] error , which is an estimator of the expected squared extraction error 1 T j T t apos;1 ( ln Ft amp;ln... In PAGE 20: ... Now we discuss the results for a smaller sample size of T = 500 observations and a larger sample size of T = 5000 observations. We show the results for T = 500 in Table 3; they are qualitatively identical to those in Table2 . Quantitatively, however, the relative performance of the quasi-maximum likelihood estimator with the log absolute return as volatility proxy, which was already poor with T = 1000 observations, is much worse with T = 500 observations.... In PAGE 20: ... D We present the results for T = 5000 in Table 4. Qualitatively, they are again identical to the results in Table2 ; quantitatively, the comparative performance of the quasi-maximum likelihood estimator with the log absolute return as volatility proxy is improved in some respects,... ..."

### Table 3. P 2 version of the central DG scheme (3.4) for the linear convection equation (3.5).

2007

"... In PAGE 9: ...4) for the linear convection equation (3.5) are listed in Table3 , with a third order TVD Runge-Kutta time discretization [45]. The results for the same equation with a = 0 are listed in Table 4, in which the rst row is computed with the previously chosen t and the second row is computed with t = x2.... In PAGE 10: ...able 7. P 2 version of the central DG scheme (3.4) for the 2D nonlinear equation. we can see that the expected third order accuracy is achieved in Table3 and fourth order accuracy, which is one order higher than expected, is achieved in Table 4. Example 2.... ..."

Cited by 2

### Table 1: The governing PDE apos;s for incompressible ow with heat transfer are shown in dimensionless form, including the Navier-Stokes equations with the Boussinesq approximation, the continuity equation, and a heat balance. Momentum

2000

"... In PAGE 5: ... In addition, a heat equation with convective and conduction terms is solved. The equations are shown in Table1 and include the time dependent terms, which are important for the formulation of the stability (eigenvalue) calcu- lation.The computational domain is discretized using a mesh of 94656 hexehe- dral elements, which corresponds to 100043 nodes.... ..."

Cited by 13

### TABLE 1 PARAMETERS OF DIFFUSION/CONVECTION MODELS

### TABLE 1 PARAMETERS OF DIFFUSION/CONVECTION MODELS

1998