A proof of the well posedness of discretized wave equation with an absorbing boundary condition

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
I. Alonso-Mallo ◽  
A.M. Portillo
Keyword(s):  
Wave Equation ◽  
Numerical Experiments ◽  
Necessary Conditions ◽  
Absorbing Boundary Conditions ◽  
Well Posedness ◽  
Initial Value ◽  
Absorbing Boundary ◽  
One Dimensional ◽  
The One ◽  
Fifth Order

Abstract- Local absorbing boundary conditions with fifth order of absorption to approximate the solution of an initial value problem, for the spatially discretized wave equation, are considered. For the one dimensional case, it is proved necessary conditions for well posedness. Numerical experiments confirming well posedness and showing good results of absorption are included.

scholarly journals Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

2011 ◽  
Vol 10 (5) ◽  
pp. 1280-1304 ◽  
Author(s):  
Pauline Klein ◽  
Xavier Antoine ◽  
Christophe Besse ◽  
Matthias Ehrhardt
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Absorbing Boundary Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Absorbing Boundary ◽  
One Dimensional ◽  
Nonlinear Potential ◽  
Linear And Nonlinear ◽  
The One

AbstractWe propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.

A symmetrical WENO-Z scheme for solving Hamilton–Jacobi equations

2020 ◽  
Vol 31 (03) ◽  
pp. 2050039
Author(s):  
Rooholah Abedian
Keyword(s):  
Viscosity Solution ◽  
Numerical Experiments ◽  
Convex Combination ◽  
One Dimensional ◽  
The One ◽  
Jacobi Equations ◽  
Fifth Order ◽  
2D And 3D ◽  
Point Stencil ◽  
The Ideal

In this paper, a new WENO procedure is proposed to approximate the viscosity solution of the Hamilton–Jacobi (HJ) equations. In the one-dimensional (1D) case, an optimum polynomial on a six-point stencil is obtained. This optimum polynomial is fifth-order accurate in regions of smoothness. Then, this optimum polynomial is considered as a symmetric and convex combination of four polynomials with ideal weights. Following the methodology of the classic WENO-Z procedure [Borges et al., J. Comput. Phys. 227, 3191 (2008)], the new nonoscillatory weights are calculated with the ideal weights. Several numerical experiments in 1D, 2D and 3D are performed to illustrate the capability of the scheme.

scholarly journals Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation

2021 ◽  
Author(s):  
Arnaud Munch ◽  
Nicolae Cindea ◽  
Arthur Bottois
Keyword(s):  
Wave Equation ◽  
Numerical Experiments ◽  
Geometric Optic ◽  
Null Controllability ◽  
Initial Condition ◽  
Observability Inequality ◽  
One Dimensional ◽  
Cylindrical Domains ◽  
The One ◽  
Uniform Observability

This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castro, C\^indea and M\"unch in SIAM J. Control Optim., 52 (2014) for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support $q$ of the control of minimal $L^2(q)$-norm. In this respect, we prove a uniform observability inequality for a class of domains $q$ satisfying the geometric optic condition. The proof based on the d'Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.

Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

2021 ◽  
Vol 130 (2) ◽  
pp. 025104
Author(s):  
Misael Ruiz-Veloz ◽  
Geminiano Martínez-Ponce ◽  
Rafael I. Fernández-Ayala ◽  
Rigoberto Castro-Beltrán ◽  
Luis Polo-Parada ◽  
...  
Keyword(s):  
Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave
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