Boundary Conditions for Constrained Systems of Evolution Equations in Numerical Relativity

2011 ◽  
Author(s):  
Tomáš Ledvinka ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Boundary Conditions ◽  
Evolution Equations ◽  
Numerical Relativity ◽  
Constrained Systems

scholarly journals Linear evolution equations on the half-line with dynamic boundary conditions

2021 ◽  
pp. 1-33
Author(s):  
D. A. SMITH ◽  
W. Y. TOH
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Evolution Equations ◽  
Half Line ◽  
Robin Problem ◽  
Dynamic Boundary Conditions ◽  
Transform Method ◽  
Linear Evolution ◽  
Robin Condition ◽  
Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

scholarly journals Constraint-preserving boundary conditions in the Z4 numerical relativity formalism

2005 ◽  
Vol 22 (13) ◽  
pp. 2615-2633 ◽  
Author(s):  
C Bona ◽  
T Ledvinka ◽  
C Palenzuela-Luque ◽  
M Žáček
Keyword(s):  
Boundary Conditions ◽  
Numerical Relativity

scholarly journals Constraint-preserving boundary conditions in numerical relativity

2002 ◽  
Vol 65 (10) ◽  
Author(s):  
Gioel Calabrese ◽  
Luis Lehner ◽  
Manuel Tiglio
Keyword(s):  
Boundary Conditions ◽  
Numerical Relativity

scholarly journals Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

2011 ◽  
Vol 152 (3) ◽  
pp. 473-496 ◽  
Author(s):  
DAVID A. SMITH
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Transform Method ◽  
Arbitrary Boundary ◽  
Initial Boundary Value ◽  
Well Posed

AbstractWe study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

Methods for determining the perturbations in VLF  in the Earth-ionosphere waveguide and TEC for ionospheric monitoring of catastrophic events in the Lithosphere-Atmosphere-Ionosphere-Magnetosphere (LAIM) system

2021 ◽  
Author(s):  
Yuriy Rapoport ◽  
Volodymyr Reshetnyk ◽  
Asen Grytsai ◽  
Alex Liashchuk ◽  
Vasyl Ivchenko ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Analytical Model ◽  
Evolution Equations ◽  
Current Source ◽  
Nonlinear Evolution Equations ◽  
Impedance Method ◽  
Wave Processes ◽  
Catastrophic Events ◽  
Impedance Boundary ◽  
The Impact

<p>A new analytical model has been developed to determine the characteristics of the VLF complex eigenmodes in the waveguide Earth-Ionosphere (WGEI). The developed analytical model, in combination with appropriate numerical methods, provides all important characteristics of (i) complex eigenmodes and their excitation by external electric current source in WGEI, (ii) radiation from WGEI of ELF waves into the upper ionosphere/magnetosphere. After propagation of electromagnetic waves over a certain distance (~ 100 km) from input antenna, either set of complex modes or input boundary conditios for an effective beams may be determined and then the theory [1] for coupled VLF beams in WGEI may be applied. This approach is considered in the context of Nonlinear evolution equations for wave processes in layered structures (NEELS) method for modeling wave processes in layered artificial/natural media [2] (i) taking into account importance of boundary conditions at the input of waveguide region, where the external current source is located, as well as complex impedance boundary conditions at WGEI boundaries [1] and (ii) using appropriate system of evolution equations for coupled wave beams. The measurements of VLF in Japan (from Hayakawa Institute of Seismo Electromagnetics) were used in this study. Fluctuations in VLF amplitudes are revealed with the periods of  (i) 2-3 hours; (ii) 5-10 minutes  corresponding to the fundamental mode of acoustic-gravity waves (AGW) near the Brunt–Väisälä period (to the best of our knowledge for the first time it has been done based on VLF signals). There is also a weekly antropogenic trend in VLF data. Since 2019, a number of VLFs and LFs signals (radiated mostly by European transmitters) have been monitored at NCCSSAU  and demonstrate the quasi-wave disturbances with periods of ~10 minutes. The analysis of the relevant data indicates the need to eliminate the impact of hardware effects. Significant influence on the distribution of TEC geomagnetic perturbations was found, in particular, during analysis of  St. Patrick's Day geomagnetic storm (March 17, 2015). To use global TEC maps for studying ionospheric processes, including possible precursors of the most powerful earthquakes, it is recommended to use data with less then one minute time resolution. Different spectral methods for data processing and entropy approach for detecting pre-catastrophic state of LAIM system are discussed. The developed methods can be useful for ionospheric monitoring of catastrophic events in the LAIM system, based on the complex studies (i) of ionospheric disturbances in VLF and TEC, in particular caused by the same source located in the lithosphere/lower atmosphere, magnetosphere or ionosphere; and (ii) the impact of AGW on VLF characteristics in WGEI.</p><p> </p><p>[1]  Rapoport Y., Grimalsky V., … Grytsai A., Fedun V. et al. Model of propagation of VLF beams in the waveguide Earth-Ionosphere. Principles of tensor impedance method in multilayered gyrotropic waveguides , Ann. Geo.,  2020,  38,  207–230.</p><p>[2] Boardman A.D., Alberucci A., … Rapoport Yu. G., … Ivchenko V.M. [et al.]. Spatial Solitonic and Nonlinear Plasmonic Aspects of Metamaterials // World Scientific Handbook of Metamaterials and Plasmonics,  2017,  4,  419-469.</p>

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