Before revision:  We adopt this methodology to derive finite-difference schemes for heat equation with Dirichlet and Neumann boundary conditions, whose convergence rates are higher than their truncation errors. We call these schemes error inhibiting schemes. When constructing a semi-discrete approximation to a partial differential equation (PDE), a discretization of the spatial operator has to be derived. For stability , this discrete operator must be semi-bounded. Under this semi-boundness, the Lax-Richtmyer equivalence theorem assures that the scheme converges at most, as the order of the error will be at most of the order of the truncation error. Usually, the error is in indeed of the order of the truncation error. In this paper, we demonstrate that schemes can be constructed such that their errors are smaller than their truncation errors. This property can enable us design schemes which are more efficient than standard schemes.
After revision:  We adopt this methodology to derive finite-difference schemes for heat equation with Dirichlet and Neumann boundary conditions, whose global error converges faster than the truncation error. Such schemes are henceforth called Error Inhibiting Schemes. We provide an explicit error analysis, including proofs of stability and convergence of the proposed schemes. We illustrate our approach using several numerical examples, which demonstrate the efficiency of our method in comparison to standard FD schemes.