scholarly journals Positivity-preserving third order DG schemes for Poisson–Nernst–Planck equations

2021 ◽  
pp. 110777
Author(s):  
Hailiang Liu ◽  
Zhongming Wang ◽  
Peimeng Yin ◽  
Hui Yu
Keyword(s):  
Third Order ◽  
Positivity Preserving

High-order finite difference WENO schemes with positivity-preserving limiter for correlated random walk with density-dependent turning rates

2015 ◽  
Vol 25 (08) ◽  
pp. 1553-1588 ◽  
Author(s):  
Yan Jiang ◽  
Chi-Wang Shu ◽  
Mengping Zhang
Keyword(s):  
Random Walk ◽  
Finite Difference ◽  
Temporal Integration ◽  
Time Discretization ◽  
High Order ◽  
Correlated Random Walk ◽  
Third Order ◽  
Positivity Preserving ◽  
Fifth Order ◽  
High Order Finite Difference

In this paper, we discuss high-order finite difference weighted essentially non-oscillatory schemes, coupled with total variation diminishing (TVD) Runge–Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintenance of third-order spatial/temporal accuracy when the limiters are applied to a third-order finite difference scheme and third-order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth-order accuracy.

A Nonoscillatory Discontinuous Galerkin Transport Scheme on the Cubed Sphere

2012 ◽  
Vol 140 (9) ◽  
pp. 3106-3126 ◽  
Author(s):  
Yifan Zhang ◽  
Ramachandran D. Nair
Keyword(s):  
Numerical Solution ◽  
Discontinuous Galerkin ◽  
Curvilinear Coordinates ◽  
Tracer Transport ◽  
Third Order ◽  
Positivity Preserving ◽  
Weighted Essentially Nonoscillatory ◽  
Sphere Geometry ◽  
Cubed Sphere ◽  
Atmospheric Tracer

Abstract The discontinuous Galerkin (DG) method is high order, conservative, and offers excellent parallel efficiency. However, when there are discontinuities in the solution, the DG transport scheme generates spurious oscillations that reduce the quality of the numerical solution. For applications such as the atmospheric tracer transport modeling, a nonoscillatory, positivity-preserving solution is a basic requirement. To suppress the oscillations in the DG solution, a limiter based on the Hermite-Weighted Essentially Nonoscillatory (H-WENO) method has been implemented for a third-order DG transport scheme. However, the H-WENO limiter can still produce wiggles with small amplitudes in the solutions, but this issue has been addressed by combining the limiter with a bound-preserving (BP) filter. The BP filter is local and easy to implement and can be used for making the solution strictly positivity preserving. The DG scheme combined with the limiter and filter preserves the accuracy of the numerical solution in the smooth regions while effectively eliminating overshoots and undershoots. The resulting nonoscillatory DG scheme is third-order accurate (P2-DG) and based on the modal discretization. The 2D Cartesian scheme is further extended to the cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates. The accuracy and effectiveness of the limiter and filter are demonstrated with several benchmark tests on both the Cartesian and spherical geometries.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

The influence of third-order interactions on the density profile of associating hard spheres

1997 ◽  
Vol 91 (4) ◽  
pp. 761-767 ◽  
Author(s):  
D. HENDERSON ◽  
S. SOKOŁOWSKI ◽  
R. ZAGORSKI ◽  
A. TROKHYMCHUK
Keyword(s):  
Density Profile ◽  
Hard Spheres ◽  
Third Order

Electro-Optic Modulator for Compensation of Third-Order Intermodulation Distortion Using Frequency Chirp Modulation

2020 ◽  
Vol E103.C (11) ◽  
pp. 653-660
Author(s):  
Daichi FURUBAYASHI ◽  
Yuta KASHIWAGI ◽  
Takanori SATO ◽  
Tadashi KAWAI ◽  
Akira ENOKIHARA ◽  
...  
Keyword(s):  
Intermodulation Distortion ◽  
Frequency Chirp ◽  
Third Order ◽  
Chirp Modulation ◽  
Electro Optic ◽  
Electro Optic Modulator
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