2.6 Gravitational waves
3.1.1 Conservation laws
In this section we consider equations in the form
∂tu+∂ifi(u) = 0, (3.1) on some domainΩwith initial datau(0, x) = u0, whereuis a vector of m unknowns,f is ad-dimensional (typically three-dimensional) flux. This is the same form of the Euler equations introduced in section2.5, cf. equation2.38, with a slightly different notation. We neglect here possible source terms on the right-hand side of3.1since they are purely algebraic and do not pose numerical issues. Note that the fluxf can be a function of the solutionubut not of its derivatives.
In general, the numerical methods discussed in this chapter can be applied to equations that are not written in the form3.1(this is the case of Einstein equations and its various formulations presented in chapter4). However many methods and techniques have been developed thinking of conservative equa-tions of this form. The reason is twofold: on one hand many physical laws can be cast in the form3.1, which transparently expresses the conservation of a quantity; on the other, when conservation laws are non-linear, numerical meth-ods based on the form3.1of the equations avoid problems known to arise in the case of discontinuous solutions. This is clearly very relevant here for the solution of the Euler equations.
A flux-conservative system of equation is said to be non-linear when the fluxf is a non-linear function ofu. The solution of such systems can develop shocks in finite time, even if the initial data is analytic. For this reason3.1has to be interpreted in the sense of distributions. A functionuis aweak solution of3.1if, for all continously differentiable test functions v(t,x)with compact support
Z ∞ 0
Z
Ω
(u∂tv+fi∂iv)dxdt= Z
Ω
u0dx . (3.2)
It can be shown however that in general even scalar conservation laws admit multiple weak solutions. To identify the “physically relevant” solution we in-troduce the concept of entropic solutions. A convex functionη(u)is said to be an entropy function if its Hessian∂u2ηsymmetrizes the Jacobian of the flux
∂uf:
∂u2η · ∂uf = [∂uf]T · ∂u2η . (3.3) If so an entropy fluxΦexists, determined by the relation
[∂uη]T · ∂uf = [∂uΦ]T. (3.4) The tuple(η,Φ)is called anentropy pair.
Anentropic solutionis weak solution that satisfies the followingentropy in-equalityfor any entropy pair
∂tη+∂iΦi(u)≤0, (3.5)
in the sense of distributions.
Kružkov(1970) proved for scalar conservation laws the existence and unique-ness of the entropic solution under very general conditions. This has been ex-tended to measure-valued solutions byDiPerna(1985), and to of conservation laws on manifolds byBen-Artzi and LeFloch(2006).
Unfortunately very little is known concerning existence, uniqueness and stability of entropic solutions in the case of systems of conservation laws, in particular for the multi-dimensional case. Not even the existence of entropy pairs is guaranteed for general systems of equations. A promising approach is the one based on divergence-measure vector fields byChen et al.(2009). In this frameworkChen and Frid(2003) proved existence, uniqueness and stability of the entropic solution of the Euler equations for a classical ideal-gas of one-dimensional Riemann problems (see the following subsection). On the other hand, for general equation of state the existence of a weak solution to the Rie-mann problem is not even guaranteedMenikoff and Plohr(1989);Chen(2006).
In the relativistic case, the existence of solutions to the Riemann problem was shown for ultrarelativistic equation of state bySmoller and Temple(1993).
The Riemann problem
TheRiemann problemfor a non-linear, hyperbolic system of conservation laws in the form (3.1) (plus possibly algebraic source terms) refers to the solution with discontinuous initial data of the form
u(0, x) =
(uL ifx <0,
uR ifx >0. (3.6)
The subscriptsLandRrefer to a “left” and “right” state of the system, respec-tively, and we have restricted the discussion to the one-dimensional case in space, since this is the most common and useful situation. This type of problem has become the standard model to study the behaviour of non-linear equations with discontinuous initial data (seeRezzolla and Zanotti(2013) for a compre-hensive introduction).
The Riemann problem also has a straightforward physical interpretation, at least when the equations under study are the hydrodynamics ones. It models a tube, filled with a fluid and divided in two halves by a membrane. The fluid in the left and right parts of the tube is in a different state of density, pressure, energy or velocity. At the initial time, the membrane is removed and the two fluids are free to interact. The evolution of the system is then given by the in-teraction of three types of non-linear hydrodynamical waves propagating from the position of the initial discontinuity: rarefaction waves, contact discontinu-ities and shocks.
The solution of the Riemann problem (which is self similar, i.e. u(t, x) = u(x/t)) cannot in general be expressed in a closed analytic form. However it can be computed to any degree of accuracy, and so it is in this sense known exactly. The solution is based on identifying the particular wave pattern for a given initial state. See figure3.1for a grphical representation of the Riemann problem solution.
The Riemann problem is relevant for the more general solution of non-linear conservation laws not only because of the theoretical insights it offers, but because many numerical schemes incorporate the solution of Riemann
Figure 3.1: Spacetime diagrams for the development of nonlinear waves in the numerical solution of a Riemann problem (the Sod problem,Sod (1978)) . Shown from the top are the rest-mass density, the pressure and the velocity.
Right panels: The corresponding profiles att = 0.25of the evolution. Figure courtesy ofRezzolla and Zanotti(2013).
problems in the solution algorithm (albeit generally employing an approximate Riemann solver rather than an exact one), starting with the classic Godunov’s method ofGodunov(1959).
There exist a number of generalizations of the Riemann problem. As a first step it is possible to consider multi-dimensional Riemann problems. The termgeneralized Riemann problemrefers however to discontinuous initial data in which the left and right states are not constant, but polynomials of higher order. Such generalizations make solving the Riemann problem considerably more difficult however.