“When we use numerical solution EXCEPT analytical solution? What are the significant of numerical solution?”
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Suppose you have a mathematical model and you want to understand its behavior. That is, you want to find a solution to the set of equations. The best is when you can use calculus, trigonometry, and other math techniques to write down the solution. Now you know absolutely how the model will behave under any circumstances. This is called the analytic solution, because you used analysis to figure it out. It is also referred to as a closed form solution.
But this tends to work only for simple models. For more complex models, the math becomes much too complicated. Then you turn to numerical methods of solving the equations, such as the Runge-Kutta method. For a differential equation that describes behavior over time, the numerical method starts with the initial values of the variables, and then uses the equations to figure out the changes in these variables over a very brief time period. Its only an approximation, but it can be a very good approximation under certain circumstances.
A computer must be used to perform the thousands of repetitive calculations involved. The result is a long list of numbers, not an equation. This long list of numbers can be used to drive an animated simulation, as we do with the models presented here.
There is also a middle ground between these two methods. There are many important non-linear equations for which it is not possible to find an analytic solution. However, there are techniques where you can find approximate analytic solutions that are close to the true solution, at least within a certain range. One such method is called the perturbation method. The advantage over a numerical solution is that you wind up with an equation (instead of just a long list of numbers) which you can gain some insight from. The Runge-Kutta algorithm is the magic formula behind most of the physics simulations shown on this web site. The Runge-Kutta algorithm lets us solve a differential equation numerically (that is, approximately); it is known to be very accurate and well-behaved for a wide range of problems.
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Imagine you are trying to solve an equation for an unknown variable, such as: x - 5 = 0. We say we have an analytic solution if we can actually solve the equation explicitly for the unknown variable. In this case, it is easy to see that the explicit analytic solution is x = 5, and that this is the exact solution. If we were not so smart, we might develop an "algorithm" on a computer to solve this equation numerically.
The algorithm would test various values for x, and then stop with a "solution" when the equation was satisfied to some chosen tolerance. For example, we might demand that the computer should solve this equation to an accuracy of 0.5. Then the computer would follow the algorithm until it found a solution to this degree of accuracy. Given an initial guess x = 1, depending on the algorithm, it might come up with the following guesses: x = 2.2 (no good), x = 3.3 (no good), x = 4.6 (good to the tolerance we specified), and return the "solution" x = 4.6! An efficient algorithm would come up with a solution quickly.
Note that if we want to be more accurate, as scientists do in their predictions, say about black holes, we might specify a tolerance that is much smaller, like 0.001. The computer might eventually get a result after the following sequence: 2.2, 3.3, 4.6, 5.2, 5.05, 4.98, 5.0005, and then return with the solution x = 5.0005. Of course, it takes much more work to solve the equation to this level of accuracy.
For such a simple equation we would never use a computer to get a solution. But the Einstein field equations contains tens of thousands of terms, and no analytic solution is possible. So we must develop algorithms to solve these equations efficiently and accurately on very large computers. The numerical solution we get in the end is essentially a collection of numbers that specify the strength of the many components of the gravitational field at various points in space, and at various times as well. This numerical solution must then be analyzed and understood through graphical representation of the data.
