اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNumerical treatment of interfaces for second-order wave equations
136
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving the second-order wave equation. We show that it is possible to implement an interface scheme of penalty type for the second-order wave equation, similar to the ones used for first-order hyperbolic and parabolic equations, and the second-order scheme used by Mattsson et al (2008). These schemes, known as SAT schemes for finite difference approximations and penalties for spectral ones, and ours share similar properties but in our case one needs to pass at the interface a smaller amount of data than previously known schemes. This is important for multi-block parallelizations in several dimensions, for it implies that one obtains the same solution quality while sharing among different computational grids only a fraction of the data one would need for a comparable (in accuracy) SAT or Mattsson et al.s scheme. The semi-discrete approximation used here preserves the norm and uses standard finite-difference operators satisfying summation by parts. For the time integrator we use a semi-implicit IMEX Runge-Kutta method. This is crucial, since the explicit Runge-Kutta method would be impractical given the severe restrictions that arise from the stiff parts of the equations.
قيم البحث
اقرأ أيضاً
In this work, we use the finite differences in time domain (FDTD) numerical method to compute and assess the validity of Hopf solutions, or hopfions, for the electromagnetic field equations. In these solutions, field lines form closed loops character
ized by different knot topologies which are preserved during their time evolution. Hopfions have been studied extensively in the past from an analytical perspective but never, to the best of our knowledge, from a numerical approach. The implementation and validation of this technique eases the study of more complex cases of this phenomena; e.g. how these fields could interact with materials (e.g. anisotropic or non-linear), their coupling with other physical systems (e.g. plasmas), and also opens the path on their artificial generation by different means (e.g. antenna arrays or lasers).
General methods of solving equations deal with solving N equations in N variables and the solutions are usually a set of discrete values. However, for problems with a softly broken symmetry these methods often first find a point which would be a solu
tion if the symmetry were exact, and is thus an approximate solution. After this, the solver needs to move in the direction of the symmetry to find the actual solution, but that can be very difficult if this direction is not a straight line in the space of variables. The solution can often be found much more quickly by adding the generators of the softly broken symmetry as auxiliary variables. This makes the number of variables more than the equations and hence there will be a family of solutions, any one of which would be acceptable. In this paper we present a procedure for finding solutions in this case, and apply it to several simple examples and an important problem in the physics of false vacuum decay. We also provide a Mathematica package that implements Powells hybrid method with the generalization to allow more variables than equations.
We study how to set the initial evolution of general cosmological fluctuations at second order, after neutrino decoupling. We compute approximate initial solutions for the transfer functions of all the relevant cosmological variables sourced by quadr
atic combinations of adiabatic and isocurvature modes. We perform these calculations in synchronous gauge, assuming a Universe described by the $Lambda$CDM model and composed of neutrinos, photons, baryons and dark matter. We highlight the importance of mixed modes, which are sourced by two different isocurvature or adiabatic modes and do not exist at the linear level. In particular, we investigate the so-called compensated isocurvature mode and find non-trivial initial evolution when it is mixed with the adiabatic mode, in contrast to the result at linear order and even at second order for the unmixed mode. Non-trivial evolution also arises when this compensated isocurvature is mixed with the neutrino density isocurvature mode. Regarding the neutrino velocity isocurvature mode, we show it unavoidably generates non-regular (decaying) modes at second order. Our results can be applied to second order Boltzmann solvers to calculate the effects of isocurvatures on non-linear observables.
We numerically solve the Klein-Gordon equation at second order in cosmological perturbation theory in closed form for a single scalar field, describing the method employed in detail. We use the slow-roll version of the second order source term and ar
gue that our method is extendable to the full equation. We consider two standard single field models and find that the results agree with previous calculations using analytic methods, where comparison is possible. Our procedure allows the evolution of second order perturbations in general and the calculation of the non-linearity parameter f_NL to be examined in cases where there is no analytical solution available.
Numerical Relativity is a mature field with many applications in Astrophysics, Cosmology and even in Fundamental Physics. As such, we are entering a stage in which new sophisticated methods adapted to open problems are being developed. In this paper,
we advocate the use of Pseudo-Spectral Collocation (PSC) methods in combination with high-order precision arithmetic for Numerical Relativity problems with high accuracy and performance requirements. The PSC method provides exponential convergence (for smooth problems, as is the case in many problems in Numerical Relativity) and we can use different bit precision without the need of changing the structure of the numerical algorithms. Moreover, the PSC method provides high-compression storage of the information. We introduce a series of techniques for combining these tools and show their potential in two problems in relativistic gravitational collapse: (i) The classical Choptuik collapse, estimating with arbitrary precision the location of the apparent horizon. (ii) Collapse in asympotically anti-de Sitter spacetimes, showing that the total energy is preserved by the numerical evolution to a very high degree of precision.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
