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.

翻译：本文对文献[Springer Proceedings in Mathematics \& Statistics, vol 237. Springer, Cham 2018 \cite{S2018}]中提出的线性方程与Hamilton-Jacobi方程耦合格式进行了全面分析。该方法融合了两种分别针对特定解特性设计的格式，为不同格式的耦合提供了通用框架，既能整合针对光滑解的精确方法，也能处理间断与梯度跃变。在\cite{S2018}中，重点是将针对间断解设计的反耗散格式与针对光滑解发展的半拉格朗日格式进行耦合。本文严格证明了所得耦合格式的基本性质，特别是在线性情形下。为验证该耦合方法的有效性，我们展示了一系列一维算例。