scholarly journals AENO: a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations

2021 ◽  
Author(s):  
Eleuterio F. Toro ◽  
Andrea Santacá ◽  
Gino I. Montecinos ◽  
Morena Celant ◽  
Lucas O. Müller
Keyword(s):  
Hyperbolic Equations ◽  
Advection Equation ◽  
Reconstruction Method ◽  
The Novel ◽  
One Dimensional ◽  
Fully Discrete ◽  
Spatial Reconstruction ◽  
Linear Advection Equation ◽  
The One ◽  
Fifth Order

AbstractIn this paper, we present a novel spatial reconstruction scheme, called AENO, that results from a special averaging of the ENO polynomial and its closest neighbour, while retaining the stencil direction decided by the ENO choice. A variant of the scheme, called m-AENO, results from averaging the modified ENO (m-ENO) polynomial and its closest neighbour. The concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system, in conjunction with the fully discrete, high-order ADER approach implemented up to fifth order of accuracy in both space and time. The results, as compared to the conventional ENO, m-ENO and WENO schemes, are very encouraging. Surprisingly, our results show that the $$L_{1}$$ L 1 -errors of the novel AENO approach are the smallest for most cases considered. Crucially, for a chosen error size, AENO turns out to be the most efficient method of all five methods tested.

The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type

2020 ◽  
Vol 28 (1) ◽  
pp. 43-52
Author(s):  
Durdimurod Kalandarovich Durdiev ◽  
Zhanna Dmitrievna Totieva
Keyword(s):  
Inverse Problem ◽  
Hyperbolic Equations ◽  
Stability Estimate ◽  
Continuous Functions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Weighted Norms ◽  
The Stability ◽  
The One ◽  
Global Unique Solvability

AbstractThe integro-differential system of viscoelasticity equations with a source of explosive type is considered. It is assumed that the coefficients of the equations depend only on one spatial variable. The problem of determining the kernel included in the integral terms of the equations is studied. The solution of the problem is reduced to one inverse problem for scalar hyperbolic equations. This inverse problem is replaced by an equivalent system of integral equations for unknown functions. The principle of constricted mapping in the space of continuous functions with weighted norms to the latter is applied. The theorem of global unique solvability is proved and the stability estimate of solution to the inverse problem is obtained.

scholarly journals FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES

2007 ◽  
Vol 17 (04) ◽  
pp. 567-591 ◽  
Author(s):  
LIVIU I. IGNAT
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Continuous Model ◽  
Local Smoothing ◽  
Euler Approximation ◽  
One Dimensional ◽  
Fully Discrete ◽  
Backward Euler ◽  
The One

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

scholarly journals A one dimensional analogue of the vorticity equation

1997 ◽  
Vol 56 (1) ◽  
pp. 119-134
Author(s):  
K. Sriskandarajah
Keyword(s):  
Soliton Solutions ◽  
Vorticity Equation ◽  
Advection Equation ◽  
Small Data ◽  
Qualitative Properties ◽  
One Dimensional ◽  
Perturbation Problem ◽  
Order Hyperbolic Equation ◽  
The One ◽  
Second Order Hyperbolic Equation

We study the qualitative properties of the one dimensional analogue of the Helmholtz vorticity advection equation. The second order hyperbolic equation has the unusual characteristic of disturbances propagating at infinite speed. The global solution for Goursat data is given in closed form. We also obtain qualitative results on the nodal curve where the solution is zero. A related perturbation problem is considered and solutions for small data are obtained. The forced vorticity equation admits a class of soliton solutions.

scholarly journals One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

2021 ◽  
Author(s):  
M. Semplice ◽  
E. Travaglia ◽  
G. Puppo
Keyword(s):  
Boundary Conditions ◽  
The Novel ◽  
Two Dimensional ◽  
Finite Volume Schemes ◽  
Third Order ◽  
One Dimensional ◽  
Numerical Tests ◽  
Ghost Cells ◽  
The One ◽  
Appl Math

AbstractWe address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252–270. 10.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells.

scholarly journals COMBINATORIAL RECONSTRUCTION OF TWO-DIMENSIONAL WORDS IN THE HYPOTHESIS SHIFT 1

2021 ◽  
Vol 1 (73) ◽  
pp. 59-61
Author(s):  
M. Ulyanov
Keyword(s):  
Linear Size ◽  
Reconstruction Method ◽  
Two Dimensional ◽  
Reconstruction Problem ◽  
Fixed Size ◽  
One Dimensional ◽  
The One

The article considers the formulation of the problem of reconstruction of two-dimensional words by a given multiset of subwords, under the hypothesis that this subset is generated by the displacement of a two-dimensional window of fixed size by an unknown two-dimensional word with a shift 1. A variant of the combinatorial solution of this reconstruction problem is proposed, based on a two-fold application of the one-dimensional word reconstruction method using the search for Eulerian paths or cycles in the de Bruyne multiorgraph. The efficiency of the method is discussed under the conditions of a square two-dimensional shift window one having a large linear size.

The Extension of Theory and Methodology to B-Splines

2009 ◽  
pp. 223-238
Author(s):  
Carlo Ciulla
Keyword(s):  
Methodological Approach ◽  
The Novel ◽  
B Spline ◽  
One Dimensional ◽  
B Splines ◽  
Sampling Locations ◽  
Starting Point ◽  
Before And After ◽  
The One

The organization of the chapter is similar to that of Chapters VII and X. The methodological approach to extend the unifying theory to the one dimensional quadratic and cubic B-Splines is herein reported along with the most relevant mathematical details. This chapter should be read along with Appendix VI where proofs are given to the assertions herein presented. In either of the two cases: quadratic and cubic B-Spline the math process starts from the calculation of the Intensity-Curvature Functional and continues with the calculation of the Sub-pixel Efficacy Region. Finally, the math process arrives to the calculation of the novel re-sampling locations through the formulas of the unifying theory seen in equations (23) and (33) for the quadratic and the cubic models respectively. The chapter concludes with a section that addresses specifically the theoretical proposition of resilient interpolation for the two classes of B-Splines. This is conducted consistently with Chapters VII and XII of the book choosing to equate the two intensity-curvature terms (before and after interpolation) as the starting point of the math deduction.

Water Quality Control of the Andelse Maas Basin, The Netherlands, by Iron Dosing

1985 ◽  
Vol 17 (4-5) ◽  
pp. 769-780 ◽  
Author(s):  
A. C. Hoekstra ◽  
K. D. Maiwald
Keyword(s):  
Chlorophyll A ◽  
Algal Growth ◽  
Phosphate Removal ◽  
Advection Equation ◽  
Flow Conditions ◽  
Water Quality Control ◽  
One Dimensional ◽  
The Hague ◽  
The One ◽  
High Phosphate

The Dune Water Works of The Hague pumps yearly about 45 million cubic meters of water from the Andelse Maas Basin to the dunes for infiltration purposes. The water in the Andelse Haas Basin is strongly influenced by the Meuse river. Phosphate concentrations are so high that algal growth causes difficulties in filters and dunes. For phosphate reduction purposes, iron is dosed into the basin. Several limnological variables are monitored based on weekly measurements. A mathematical model has been developed for the description of the eutrophication in the basin. The model is time and space dependent and based on the one-dimensional dispersion-advection equation. Two separate submodels are used, one for the simulation of flow conditions and one for the algae and nutrient kinetics. The ortho-phosphate reduction is modelled as a first order process. The eutrophication model has been calibrated, verified and used for a prediction of chlorophyll-a and ortho-phosphate concentrations for the case of the shut down of the iron dosing installation. The dosing of iron appears to be a very effective way for phosphate removal. Termination of the dosing would give unacceptable high phosphate and chlorophyll-a concentrations. Present investigations are focussed upon the minimization of the amount of iron dosing.

SUBCRITICAL TRANSITION TOWARDS SPACE-TIME CHAOS

2001 ◽  
Vol 11 (11) ◽  
pp. 2809-2816 ◽  
Author(s):  
L. PASTUR ◽  
R. RIBOTTA
Keyword(s):  
Landau Equation ◽  
Model System ◽  
Space Time ◽  
Selection Mechanism ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Localized State ◽  
The One ◽  
Fifth Order ◽  
Experimental Model System

The subcritical transition towards space-time chaos of a nonlinear localized state (defined by well-defined fronts) is studied along the bifurcated hysteretic branch of the bifurcation. The experimental model system is the one-dimensional electroconvection of a nematic liquid crystal. The transition reveals an original cascade of bifurcations between front regimes, as well as a width-selection mechanism for the domains. A fifth-order Ginzburg–Landau equation reproduces most of the experimental observations.

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