An Introduction to Well-Posedness and Free-Evolution

Abstract

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace-Fourier method for analyzing the initial boundary value problem. Finally I state how these notions extend to systems that are first order in time and second order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Exploring New Physics Frontiers Through Numerical Relativity

“…This aspect is referred to as well posedness of the IBVP and is discussed in great detail in Living Reviews articles and other works [645, 674, 383, 427]. Here, we merely list the basic concepts and refer the interested reader to these articles.…”

Numerical Relativity in Higher-Dimensional Space–times

“…(3), the fact that the induced metric is spatial, i.e., γ M N n N = 0 and the definition of the extrinsic curvature (12). If we now insert Eq.…”

“…As we have noted before and is discussed in detail in Hilditch's contibution to the lecture notes [12] this formulation is prone to numerical instabilities due to its PDE structure. To "cure" this instability we need to re-write Eqs.…”

“…As Hilditch discusses in great detail in his contribution to these lecture notes [12], a strongly hyperbolic initial value formulation of the PDE system is a necessary condition to obtain a stable numerical scheme. By now, there is a plethora of well-posed initial value formulations of Einstein's equations.…”

“…A necessary condition for numerical stability is the well-posedness of the continuum PDE system as is discussed in Hilditch's contribution to these lecture notes [12]. Typically, NR methods imply heavy numerical simulations in 3 + 1-dimensional setups.…”