 Solution

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 $k\downarrow 0$; 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.