Perhaps this requires a more advanced knowledge of differential equations, but are solutions in numerical relativity (for example, the merger of two black holes) actual solutions to the Einstein Field Equations? Or rather, are they approximations to what the Einstein Field Equations would yield? I have generally taken "numerical solution" to mean an approximation to the solution of a differential equation, as in the latter case.
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3$\begingroup$ Do only exact solutions count as “actual” solutions? What if an approximate solution can in principle be made arbitrarily close to exact with enough computational power? $\endgroup$– G. SmithCommented Apr 3, 2020 at 4:19
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$\begingroup$ 3.1415 is only an approximation to $\pi$, it is nevertheless very useful for practical purposes $\endgroup$– Prof. LegolasovCommented Apr 3, 2020 at 7:57
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That depends on what you mean by "actual solution". By their nature numerical solutions are approximations to exact solutions. Approximations come in at various stages in numerical calculations, for example in choices of grid size, truncations of infinite series, or even just the fact that computer arithmetic has only finite precision.
However, by construction a numerical solution will be close to an exact solutions, and good numerical methods will indicate how "far away" the obtained result is from the exact solution. And ideally, underlying the numerical method there will be a proof that the numerical method will converge to an exact solution (given enough precision, small enough grid size, etc.).
