The Kirchhoff transformation is an effective means for dealing with temperature dependent conductivities. In general numerical applications, however, the use of this approach will produce non-linear discrete equations, which can be costly to solve. This paper introduces a local Kirchhoff approach for approximating the conductivity terms in the discrete equation. This approach results in an efficient solution in terms of temperature alone. Application to a problem with rapidly changing conductivity shows that use of high-order numerical integration in the conductivity approximation leads to very accurate predictions.