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Typically when a semi-discrete approximation to a partial differentialequation (PDE) is constructed a discretization of the spatial operator with atruncation error $\tau$ is derived. This discrete operator should besemi-bounded for the scheme to be stable. Under these conditions, theLax--Ricchtmyer equivalence theorem assures that the scheme converges and thatthe error will be, at most, of the order of $\| \tau \|$. In most cases, theerror is in indeed of the order of $\| \tau \|$.  We demonstrate that for the Heat equation stable schemes can be constructed,whose truncation errors are $\tau$, however, the actual errors are muchsmaller. This gives more degrees of freedom in the design of schemes which canmake them more efficient (more accurate or compact) than standard schemes. Insome cases, the accuracy of the schemes can be further enhanced usingpost-processing procedures.

High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors