scholarly journals A Third Order Accurate Cellwise Relaxation Implicit Discontinuous Galerkin Scheme for Unstructured Hybrid Meshes

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
Hiroyuki Asada ◽  
Yousuke Ogino ◽  
Kanako Yasue ◽  
Keisuke Sawada
Keyword(s):  
Parabolic Equation ◽  
Discontinuous Galerkin ◽  
Computational Cost ◽  
Convergence Properties ◽  
Third Order ◽  
The Third ◽  
Discontinuous Galerkin Scheme ◽  
Aerospace Applications ◽  
Hybrid Meshes ◽  
Rans Simulations

A third order accurate cellwise relaxation implicit Discontinuous Galerkin (DG) scheme for RANS simulations using unstructured hybrid meshes is presented. A scalar parabolic equation is first examined to clarify what is really important in construction of implicit matrix to keep its diagonal dominance for the third and fourth order cellwise relaxation implicit DG schemes. In addition, discussions are given to approximated construction of implicit matrix for reducing computational cost. Then, the third order accurate cellwise relaxation implicit DG scheme for RANS simulations is successfully developed utilizing the expertise learned in the study of solving the parabolic equation. Superior spatial accuracy of the third order accurate cellwise relaxation implicit DG scheme for RANS simulations, while retaining reasonable convergence properties, is demonstrated for typical aerospace applications.

scholarly journals Implementable tensor methods in unconstrained convex optimization

2019 ◽  
Author(s):  
Yurii Nesterov
Keyword(s):  
Convex Optimization ◽  
Auxiliary Problem ◽  
Computational Cost ◽  
Rates Of Convergence ◽  
Smoothness Condition ◽  
Third Order ◽  
Worst Case ◽  
Complexity Bounds ◽  
The Third ◽  
Tensor Methods

AbstractIn this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level $$O\left( {1 \over k^4}\right) $$O1k4, where k is the number of iterations. This is very close to the lower bound of the order $$O\left( {1 \over k^5}\right) $$O1k5, which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.

scholarly journals Combined DG scheme that maintains increased accuracy in areas of shock waves

2019 ◽  
Vol 489 (2) ◽  
pp. 119-124 ◽  
Author(s):  
M. E. Ladonkina ◽  
O. A. Nekliudova ◽  
V. V. Ostapenko ◽  
V.  F. Tishkin
Keyword(s):  
Shock Waves ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Nonlinear Correction ◽  
Third Order ◽  
The Third ◽  
Dg Method

A combined scheme of the discontinuous Galerkin method is proposed. This scheme monotonously localizes the fronts of shock waves and simultaneously maintains increased accuracy in the regions of smoothness of the calculated solutions. In this scheme, a non-monotonic version of the third-order DG method is used as the baseline and a monotonic version of this method is used as the internal one, in which a nonlinear correction of numerical flows is used. Tests demonstrating the advantages of the new scheme compared to the standard monotonized variants of the DG method are provided.

scholarly journals A simple and accurate discontinuous Galerkin scheme for modeling scalar-wave propagation in media with curved interfaces

Geophysics ◽  
2015 ◽  
Vol 80 (2) ◽  
pp. T83-T89 ◽  
Author(s):  
Xiangxiong Zhang ◽  
Sirui Tan
Keyword(s):  
Discontinuous Galerkin ◽  
Large Scale ◽  
Computational Cost ◽  
Scalar Wave ◽  
Material Interfaces ◽  
Galerkin Scheme ◽  
Discontinuous Galerkin Scheme ◽  
Normal Vectors ◽  
Second Order Convergence ◽  
Conventional Scheme

Conventional high-order discontinuous Galerkin (DG) schemes suffer from interface errors caused by the misalignment between straight-sided elements and curved material interfaces. We have developed a novel DG scheme to reduce those errors. Our new scheme uses the correct normal vectors to the curved interfaces, whereas the conventional scheme uses the normal vectors to the element edge. We modify the numerical fluxes to account for the curved interface. Our numerical modeling examples demonstrate that our new discontinuous Galerkin scheme gives errors with much smaller magnitudes compared with the conventional scheme, although both schemes have second-order convergence. Moreover, our method significantly suppresses the spurious diffractions seen in the results obtained using the conventional scheme. The computational cost of our scheme is similar to that of the conventional scheme. The new DG scheme we developed is, thus, particularly useful for large-scale scalar-wave modeling involving complex subsurface structures.

scholarly journals NUMERICAL EXPLORATION OF A FORWARD–BACKWARD DIFFUSION EQUATION

2012 ◽  
Vol 22 (06) ◽  
pp. 1250004 ◽  
Author(s):  
P. LAFITTE ◽  
C. MASCIA
Keyword(s):  
Parabolic Equation ◽  
Diffusion Equation ◽  
Singular Limit ◽  
Third Order ◽  
The Third ◽  
Numerical Exploration ◽  
Main Emphasis ◽  
Analytical Properties ◽  
Diffusion Function ◽  
Entropy Formulation

We analyze numerically a forward–backward diffusion equation with a cubic-like diffusion function — emerging in the framework of phase transitions modeling — and its "entropy" formulation determined by considering it as the singular limit of a third-order pseudo-parabolic equation. Precisely, we propose schemes for both the second- and the third-order equations, we discuss the analytical properties of their semi-discrete counterparts and we compare the numerical results in the case of initial data of Riemann type, showing strengths and flaws of the two approaches, the main emphasis being given to the propagation of transition interfaces.

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.

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