Energy Stable WENO Schemes of Arbitrary Order

2011 ◽  
pp. 81-87
Author(s):  
Nail K. Yamaleev ◽  
M.H. Carpenter
Keyword(s):  
Arbitrary Order ◽  
Weno Schemes ◽  
Energy Stable

scholarly journals A systematic methodology for constructing high-order energy stable WENO schemes

2009 ◽  
Vol 228 (11) ◽  
pp. 4248-4272 ◽  
Author(s):  
Nail K. Yamaleev ◽  
Mark H. Carpenter
Keyword(s):  
High Order ◽  
Weno Schemes ◽  
Order Energy ◽  
Systematic Methodology ◽  
Energy Stable

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Acta Numerica ◽  
2020 ◽  
Vol 29 ◽  
pp. 701-762
Author(s):  
Chi-Wang Shu
Keyword(s):  
Main Idea ◽  
High Order Accuracy ◽  
Finite Difference Schemes ◽  
Approximation Procedure ◽  
Discontinuous Solutions ◽  
Weno Schemes ◽  
Convection Diffusion ◽  
Different Types ◽  
Basic Ideas ◽  
Weighted Eno

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

scholarly journals The determinant of one-dimensional polyharmonic operators of arbitrary order

2020 ◽  
Vol 279 (12) ◽  
pp. 108783
Author(s):  
Pedro Freitas ◽  
Jiří Lipovský
Keyword(s):  
Arbitrary Order ◽  
One Dimensional

scholarly journals Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
Elena Bachini ◽  
Gianmarco Manzini ◽  
Mario Putti
Keyword(s):  
Arbitrary Order ◽  
Surface Geometry ◽  
Virtual Element Method ◽  
Anisotropic Character ◽  
Local Reference System ◽  
Local Reference ◽  
Local Parametrization ◽  
Manufactured Solution ◽  
Virtual Element ◽  
Element Method

AbstractWe develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

scholarly journals Polyanalytic boundary value problems for planar domains with harmonic Green function

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Heinrich Begehr ◽  
Bibinur Shupeyeva
Keyword(s):  
Green Function ◽  
Boundary Value Problems ◽  
Arbitrary Order ◽  
Boundary Value ◽  
Planar Domains ◽  
Schwarz Problem ◽  
Bianalytic Functions ◽  
Neumann Type ◽  
The Neumann Problem ◽  
Well Posed

AbstractThere are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

scholarly journals Horndeski genesis: consistency of classical theory

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yulia Ageeva ◽  
Pavel Petrov ◽  
Valery Rubakov
Keyword(s):  
Strong Coupling ◽  
Dimensional Analysis ◽  
Classical Theory ◽  
Arbitrary Order ◽  
Specific Model ◽  
Power Counting ◽  
Coupling Energy ◽  
Classical Description ◽  
Higher Order Terms ◽  
The Universe

Abstract Genesis within the Horndeski theory is one of possible scenarios for the start of the Universe. In this model, the absence of instabilities is obtained at the expense of the property that coefficients, serving as effective Planck masses, vanish in the asymptotics t → −∞, which signalizes the danger of strong coupling and inconsistency of the classical treatment. We investigate this problem in a specific model and extend the analysis of cubic action for perturbations (arXiv:2003.01202) to arbitrary order. Our study is based on power counting and dimensional analysis of the higher order terms. We derive the latter, find characteristic strong coupling energy scales and obtain the conditions for the validity of the classical description. Curiously, we find that the strongest condition is the same as that obtained in already examined cubic case.

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