The paper presents a class of new explicit second-order accurate finite difference schemes for computing weak solutions of hyperbolic conservation laws. These schemes are derived by applying a nonoscillatory first-order accurate scheme to a modified flux function. The resulting second-order schemes achieve high resolution while preserving the robustness of the original nonoscillatory scheme. The authors demonstrate the performance of these schemes through numerical experiments. The introduction covers the background of hyperbolic systems of conservation laws, including the definition of entropy functions and the conditions for convergence of finite-difference solutions. The paper then discusses monotonicity properties in the scalar case and the conditions for total variation non-increasing (TVNI) schemes. It introduces a hierarchy of properties, including monotonicity preservation and TVNI, and provides a proof for these properties. The main section of the paper describes the construction of second-order accurate TVNI schemes. It starts with a 3-point first-order accurate TVNI scheme and transforms it into a 5-point second-order accurate TVNI scheme by modifying the numerical flux. The authors provide a heuristic argument and detailed mathematical proofs to support their construction. The paper also extends the scalar scheme to systems of conservation laws using a generalized version of Roe's technique. This extension involves applying the scalar scheme to each characteristic variable of the system. The authors prove that the resulting scheme is TVNI under certain CFL restrictions and demonstrate its convergence for initial data of bounded total variation. Finally, the paper discusses the relationship between entropy enforcement and resolution, providing numerical experiments and a heuristic analysis to explain the observed performance of the schemes. The authors conclude by discussing the challenges and potential improvements in the context of linearly degenerate fields.The paper presents a class of new explicit second-order accurate finite difference schemes for computing weak solutions of hyperbolic conservation laws. These schemes are derived by applying a nonoscillatory first-order accurate scheme to a modified flux function. The resulting second-order schemes achieve high resolution while preserving the robustness of the original nonoscillatory scheme. The authors demonstrate the performance of these schemes through numerical experiments. The introduction covers the background of hyperbolic systems of conservation laws, including the definition of entropy functions and the conditions for convergence of finite-difference solutions. The paper then discusses monotonicity properties in the scalar case and the conditions for total variation non-increasing (TVNI) schemes. It introduces a hierarchy of properties, including monotonicity preservation and TVNI, and provides a proof for these properties. The main section of the paper describes the construction of second-order accurate TVNI schemes. It starts with a 3-point first-order accurate TVNI scheme and transforms it into a 5-point second-order accurate TVNI scheme by modifying the numerical flux. The authors provide a heuristic argument and detailed mathematical proofs to support their construction. The paper also extends the scalar scheme to systems of conservation laws using a generalized version of Roe's technique. This extension involves applying the scalar scheme to each characteristic variable of the system. The authors prove that the resulting scheme is TVNI under certain CFL restrictions and demonstrate its convergence for initial data of bounded total variation. Finally, the paper discusses the relationship between entropy enforcement and resolution, providing numerical experiments and a heuristic analysis to explain the observed performance of the schemes. The authors conclude by discussing the challenges and potential improvements in the context of linearly degenerate fields.