Iterative methods - causality violation?

[sri sharan]

Consider a partial differential equation describing the evolution of some function of a system which varies with time and space. A set of initial conditions and boundary are absolutely necessary for solving the equation. However, there are some numerical iterative methods for solving differential equations like Gauss sidel, where any arbitrary initial condition can be assumed. And yet it yields an accurate solution(with the error of approximation present obviously). So how is it that the numerical technique, which represents the differential equation was able to solve without initial conditions? How can a solution be generated without talking the initial information which was essential for the original differential equation?

Edit: I realize that the title isn't exactly apt. However I can't change it now, so sorry about that

 

[gtoguy97]

.The numerical iterative methods like Gauss Siedel are algorithms that approximate the solution to the differential equation. The initial conditions and boundary conditions are used to define the system and its boundaries, but they are not required for the algorithm itself. The numerical algorithms are designed to take an arbitrary set of data, and then iteratively refine the data until it converges on a close approximation of the solution. This means that the initial conditions don't have to be exact or even close to the actual solution, because the algorithm will refine the data until it finds a close approximation. So while the initial conditions are necessary for solving the differential equation, the numerical algorithms don't need them in order to find an approximate solution.