  We present a comprehensive analysis of the coupled scheme introduced in
[Springer Proceedings in Mathematics \& Statistics, vol 237. Springer, Cham
2018 \cite{S2018}] for linear and Hamilton-Jacobi equations. This method merges
two distinct schemes, each tailored to handle specific solution
characteristics. It offers a versatile framework for coupling various schemes,
enabling the integration of accurate methods for smooth solutions and the
treatment of discontinuities and gradient jumps. In \cite{S2018}, the emphasis
was on coupling an anti-dissipative scheme designed for discontinuous solutions
with a semi-Lagrangian scheme developed for smooth solutions. In this paper, we
rigorously establish the essential properties of the resulting coupled scheme,
especially in the linear case. To illustrate the effectiveness of this coupled
approach, we present a series of one-dimensional examples.
