Space discretisation errors

Spatial discretisation introduces further errors. From the Taylor expansion in space, the local error for one step in time due to spatial discretisation is proportional to $\Delta x^2\Delta t$, which means that the accumulated errors should be approximately proportional to $\Delta x^2 = \Delta x^2\Delta t/\Delta t$. Again, this can be reduced by halving the step-size in space, until the changes that occur are less than the error tolerance.