13 XI. EVOLUTION EQUATIONS FOR A NUMERICAL TREATMENT. A. Quasi–local evolution equations. which yields L$_{j}$ by integration (as m$_{i}$ and m$_{j}$ are known and R$_{i}$ can be fixed as a coordinate gauge). It is straight- forward to show that the analytic solutions for the hy- perbolic class yield a unique LTB model mapping these profiles if (97) holds. Krasi´ nski and Hellaby also examined the mapping of profiles of velocities defined as v ≡ ˙ R/R$_{i}$ = L H $_{q}$. To deal with this case we eliminate k$_{qi}$ as k$_{qi}$ = 2 m$_{qi}$ − H 2 $_{qi}$, so that the initial value functions are m$_{qi}$ and H $_{qi}$. Then, from the Hamiltonian constraint (38) we obtain The analytic solutions, either in terms of the con- ventional variables M, E, ct bb , R or the initial value parametrization m$_{qi}$, k$_{qi}$, R$_{i}$, L that we have presented, are mostly useful for qualitative work (as illustrated in the previous section). We remark that these solutions are either parametric or implicit, and thus a pure ana- lytic framework based on them is necessary limited. For concrete applications in models of less idealized cosmo- logical inhomogeneities and observations there is no other way but to consider a numeric framework. The 1+3 fluid flow system (18)–(20) discussed in sec- tion IV provides a fully covariant description and can be used to determine the dynamics of LTB models by means of numeric techniques. However, the associated spacelike constraints (19) are partial differential equation on r that are not easy to solve, making it difficult (in general) to decouple the time and radial derivatives. As we show in this section, the quasi–local scalars and their fluctuations (in the initial value parametrization) provide more con- venient variables for a numeric treatment of LTB models. An alternative system to the evolution equations (18)– (20) follows by considering a 1+3 fluid flow evolu- tion equations for the quasi–local scalars (35), involv- ing m$_{q}$, H q and their corresponding relative fluctuations δ ( m $^{)}$, δ( $^{)}$. This system can be obtained by eliminating ρ, Θ , Σ and E in (18) in terms of the new variables by means of (32) and using (31) to deal with the constraints (19). Considering the notation (29): H = Θ / 3 , m = κρ/ 3 and k = $^{3}$R / 6, this framework leads to the follow- ing evolution equations while the spacelike constrains (19) become the equations defining δ ( m ) and δ ( H ) through (27a) and (31), and the Hamiltonian constraint (20) reduces to (38). This is the particular case of pure dust in [30, 31]. The system (102) is equivalent to (18), but it is much more practical and easier to use in a numerical treat- ment, as it is not necessary to solve any radial differen- tial equation as a precondition to solve it. In practice, we can handle it as a system of non–linear autonomous ordinary differential equations, where r enters as a pa- rameter (see Appendix B of [29]). Notice that once we have solved (102) all local scalars in (17) can be deter- mined by means of (32), (38) and (43). The issue of mapping two profiles can also be under- stood in terms of the systems of evolution equations that we derive in the following section. These systems are partial differential equations, hence initial and boundary conditions are given as functions of r (or radial profiles). The selection of two initial value functions is sufficient to determine a unique solution. These functions can be, either m$_{qi}$, H qi or m$_{qi}$, k$_{qi}$ or ˆ Ω$_{i}$ , H $_{qi}$. Hence if we know only one of these functions (evaluated at t = t$_{i}$ ) and the same function at t = t$_{j}$ , then we may use one of the avail- able constraints to find the second initial value function, and thus guarantee (if the constraint has a solution) that the evolution system yields a unique LTB model. We have proceeded in this section (following [7, 10]) by us- ing the analytic solutions as the constraints that link the boundary and initial conditions through L . The results are the same as those of [7, 10], but the initial value parametrization of the solutions makes the whole proce- dure a lot more natural and straightforward. where W and W$_{i}$ take the forms (84a)–(84b) or (85a)– (85b) with L given by (100) for hyperbolic or elliptic models or regions. Evidently, obtaining the missing ini- tial value function by means of constraints like (93) and (101) does not guarantee the fulfillment of regularity con- ditions (like absence of shell crossings). This has to be verified independently by means of the Hellaby–Lake con- ditions once the LTB model mapping the profiles has been found. which substituted in the solutions (82) and (83) yields the missing initial value function H qi by means of the constraint which inserted in the solutions (50) and (55) yields the constraints needed to obtain the missing initial value function m$_{qi}$ . We can also obtain the LTB model that results from assuming that the profiles ˆ Ω$_{i}$ and ˆ Ω$_{j}$ are known. In this case we obtain L$_{j}$ from (77) L$_{j}$ = 2 m$_{qi}$R 2 i v 2 j − v 2 i + 2 m$_{qi}$R 2 i , (99) L$_{j}$ = ˆ Ω$_{i}$ ˆ Ω$_{j}$ 1 − ˆ Ω$_{j}$ 1 ˆ Ω$_{i}$ , (100) H qi = W ( ˆ Ω$_{i}$ , ˆ Ω$_{j}$) − W$_{i}$ ( ˆ Ω$_{i}$) c ( t$_{j}$ t$_{i}$ ) , (101) ˙ m$_{q}$ = − 3 m$_{q}$ H $_{q}$, (102a) ˙ H q = −H 2 q − m$_{q}$, (102b) ˙ δ ( m ) = − 3(1 + δ ( m $^{)}$) H $_{q}$δ ( H $^{)}$, (102c) ˙ δ ( H ) = − (1 + δ ( H $^{)}$) H $_{q}$δ ( H ) + m$_{q}$ H q ( δ ( H ) − δ ( m $^{)}$) , (102d)