Correction Procedure via Reconstruction Using Summation-by-Parts Operators

2018 ◽  
pp. 491-501
Author(s):  
Philipp Öffner ◽  
Hendrik Ranocha ◽  
Thomas Sonar
Keyword(s):  
Correction Procedure ◽  
Summation By Parts

scholarly journals Summation-by-parts operators for correction procedure via reconstruction

2016 ◽  
Vol 311 ◽  
pp. 299-328 ◽  
Author(s):  
Hendrik Ranocha ◽  
Philipp Öffner ◽  
Thomas Sonar
Keyword(s):  
Correction Procedure ◽  
Summation By Parts

scholarly journals Stability of correction procedure via reconstruction with summation-by-parts operators for Burgers' equation using a polynomial chaos approach

2018 ◽  
Vol 52 (6) ◽  
pp. 2215-2245 ◽  
Author(s):  
Philipp Öffner ◽  
Jan Glaubitz ◽  
Hendrik Ranocha
Keyword(s):  
Polynomial Chaos ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Conditions ◽  
High Sensitivity ◽  
Correction Procedure ◽  
Summation By Parts ◽  
Numerical Fluxes ◽  
Entropy Stable ◽  
The Given

In this paper, we consider Burgers’ equation with uncertain boundary and initial conditions. The polynomial chaos (PC) approach yields a hyperbolic system of deterministic equations, which can be solved by several numerical methods. Here, we apply the correction procedure via reconstruction (CPR) using summation-by-parts operators. We focus especially on stability, which is proven for CPR methods and the systems arising from the PC approach. Due to the usage of split-forms, the major challenge is to construct entropy stable numerical fluxes. For the first time, such numerical fluxes are constructed for all systems resulting from the PC approach for Burgers' equation. In numerical tests, we verify our results and show also the performance of the given ansatz using CPR methods. Moreover, one of the simulations, i.e. Burgers’ equation equipped with an initial shock, demonstrates quite fascinating observations. The behaviour of the numerical solutions from several methods (finite volume, finite difference, CPR) differ significantly from each other. Through careful investigations, we conclude that the reason for this is the high sensitivity of the system to varying dissipation. Furthermore, it should be stressed that the system is not strictly hyperbolic with genuinely nonlinear or linearly degenerate fields.

Lateral resolution of the electron microbeam analysis

1978 ◽  
Vol 36 (1) ◽  
pp. 542-543
Author(s):  
H.J. Dudek
Keyword(s):  
Quantitative Analysis ◽  
Depth Resolution ◽  
Lateral Resolution ◽  
Cylindrical Particle ◽  
Correction Procedure ◽  
X Ray ◽  
Element Concentrations ◽  
Microbeam Analysis ◽  
The Matrix

The chemical inhomogenities in modern materials such as fibers, phases and inclusions, often have diameters in the region of one micrometer. Using electron microbeam analysis for the determination of the element concentrations one has to know the smallest possible diameter of such regions for a given accuracy of the quantitative analysis.In th is paper the correction procedure for the quantitative electron microbeam analysis is extended to a spacial problem to determine the smallest possible measurements of a cylindrical particle P of high D (depth resolution) and diameter L (lateral resolution) embeded in a matrix M and which has to be analysed quantitative with the accuracy q. The mathematical accounts lead to the following form of the characteristic x-ray intens ity of the element i of a particle P embeded in the matrix M in relation to the intensity of a standard S

Bence-albee after 25 years: A new superior α-factor correction procedure for the quantitative analysis of oxides and silicates

1992 ◽  
Vol 50 (2) ◽  
pp. 1744-1745
Author(s):  
John T. Armstrong
Keyword(s):  
Quantitative Analysis ◽  
Electron Microprobe ◽  
Binary Systems ◽  
Correction Procedure ◽  
Geological Samples ◽  
The Past ◽  
Large Matrix ◽  
Computational Speed ◽  
Microprobe Data ◽  
Geological Sciences

One of the most cited papers in the geological sciences has been that of Albee and Bence on the use of empirical " α -factors" to correct quantitative electron microprobe data. During the past 25 years this method has remained the most commonly used correction for geological samples, despite the facts that few investigators have actually determined empirical α-factors, but instead employ tables of calculated α-factors using one of the conventional "ZAF" correction programs; a number of investigators have shown that the assumption that an α-factor is constant in binary systems where there are large matrix corrections is incorrect (e.g, 2-3); and the procedure’s desirability in terms of program size and computational speed is much less important today because of developments in computing capabilities. The question thus exists whether it is time to honorably retire the Bence-Albee procedure and turn to more modern, robust correction methods. This paper proposes that, although it is perhaps time to retire the original Bence-Albee procedure, it should be replaced by a similar method based on compositiondependent polynomial α-factor expressions.

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