High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors
This work provides a new design principle for finite difference schemes that can yield higher accuracy than traditional error analysis suggests, benefiting computational scientists solving the heat equation.
The authors construct stable finite difference schemes for the heat equation whose actual convergence rates are higher than their truncation errors, leading to more efficient schemes. For example, they achieve errors much smaller than the truncation error order, and post-processing can further enhance accuracy.
Typically when a semi-discrete approximation to a partial differential equation (PDE) is constructed a discretization of the spatial operator with a truncation error $τ$ is derived. This discrete operator should be semi-bounded for the scheme to be stable. Under these conditions, the Lax--Ricchtmyer equivalence theorem assures that the scheme converges and that the error will be, at most, of the order of $\| τ\|$. In most cases, the error is in indeed of the order of $\| τ\|$. We demonstrate that for the Heat equation stable schemes can be constructed, whose truncation errors are $τ$, however, the actual errors are much smaller. This gives more degrees of freedom in the design of schemes which can make them more efficient (more accurate or compact) than standard schemes. In some cases, the accuracy of the schemes can be further enhanced using post-processing procedures.