consistency
When you replace a PDE by a finite-difference scheme on a grid, you are obviously approximating — but you want a basic guarantee: that as the grid spacing shrinks toward zero, the difference equation you are actually solving really does turn back into the PDE you meant to solve, and not into some other equation. Consistency is that minimum sanity check. A scheme that is NOT consistent is solving the wrong problem no matter how fine you make the grid.
Precisely: take the exact solution of the PDE and plug it into your difference scheme. It will not satisfy the scheme exactly — there is a leftover, the LOCAL TRUNCATION ERROR, which measures by how much the true solution fails to obey the discrete equation. The scheme is consistent if this local truncation error tends to zero as h and k go to zero. You compute it with Taylor series: expand each grid value about a central point, and the PDE's terms cancel, leaving error terms like (k/2) u_tt - (alpha h^2 / 12) u_xxxx + .... The lowest powers of h and k that survive give the ORDER of the scheme — for example O(k + h^2) for FTCS, O(k^2 + h^2) for Crank-Nicolson.
Consistency is necessary but, on its own, almost worthless — it is a statement about a single step, not about whether errors accumulate over many steps. A scheme can be perfectly consistent yet UNSTABLE, so that tiny per-step errors are amplified into garbage. That is why consistency is only one leg of the tripod: the Lax equivalence theorem says that for a consistent linear scheme, CONVERGENCE (the numerical solution approaching the true one) holds if and only if the scheme is also STABLE. Consistency tells you the target is right; stability tells you that you will actually hit it.
Plug a smooth exact solution into FTCS for u_t = alpha*u_xx. Taylor expansion shows the scheme actually equals u_t - alpha*u_xx + (k/2)u_tt - (alpha h^2/12)u_xxxx + .... The PDE part is zero, leaving local truncation error O(k) + O(h^2), which vanishes as k, h -> 0. FTCS is therefore consistent, of order one in time and two in space.
Plug the true solution in; if the leftover vanishes as the grid shrinks, the scheme is consistent.
Consistency is about a SINGLE step's local truncation error, not the final answer. A consistent scheme can still converge to the wrong thing or blow up if it is unstable — consistency alone never implies convergence. Watch too for tricky cases (like the DuFort-Frankel scheme) where consistency depends on HOW k and h shrink together.