= Solution
\b[Refining time alone converges to the fixed-grid, spatially discretized solution], assuming the time-dependent coefficients and boundary forcing are suitably regular. In time-to-maturity $\tau=T-t$, the grid values solve a finite system $dU/d\tau=A_h(\tau)U+F_h(\tau)$. For fixed $h$, the <explicit Euler method> converges to this system as $k\downarrow0$; time-step error decreases to zero.
The spatial <finite difference> error remains. Thus the limit is not in general the exact continuous pricing solution: sufficiently small time steps reveal an error plateau of order $h^2$ under the usual smooth-solution assumptions. A truncated-domain boundary error also remains fixed unless negligible as stipulated.
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