This 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 an appropriately modified flux function. The resulting second-order schemes achieve high resolution while preserving the robustness of the original first-order scheme. The schemes are designed to be total variation nonincreasing (TVNI), ensuring convergence to weak solutions that satisfy entropy conditions. The paper discusses the convergence of finite-difference schemes to weak solutions of hyperbolic conservation laws. It shows that under certain conditions, such as bounded total variation and consistency with entropy conditions, the finite-difference solutions converge to weak solutions. The paper also addresses the issue of artificial viscosity, which can introduce information loss but is used to simulate the zero-dissipation limit for selecting physically relevant solutions. In the scalar case, the paper analyzes monotonicity and TVNI properties of finite-difference schemes. It shows that monotone schemes are TVNI and that TVNI schemes are monotonicity preserving. The paper then describes how to convert first-order accurate TVNI schemes into second-order accurate TVNI schemes by modifying the flux function and introducing a numerical viscosity term. For systems of conservation laws, the paper extends the scalar scheme by applying it to each characteristic field. The resulting schemes are TVNI and second-order accurate. The paper also discusses the role of the function Q(x) in determining the numerical viscosity and the resolution of shocks and contact discontinuities. The paper concludes with a discussion of entropy enforcement and resolution in both genuinely nonlinear and linearly degenerate fields. It shows that the proposed schemes achieve high resolution and strong entropy enforcement, even in the presence of contact discontinuities. The paper also addresses the issue of numerical viscosity and its impact on the resolution of shocks and contact discontinuities.This 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 an appropriately modified flux function. The resulting second-order schemes achieve high resolution while preserving the robustness of the original first-order scheme. The schemes are designed to be total variation nonincreasing (TVNI), ensuring convergence to weak solutions that satisfy entropy conditions. The paper discusses the convergence of finite-difference schemes to weak solutions of hyperbolic conservation laws. It shows that under certain conditions, such as bounded total variation and consistency with entropy conditions, the finite-difference solutions converge to weak solutions. The paper also addresses the issue of artificial viscosity, which can introduce information loss but is used to simulate the zero-dissipation limit for selecting physically relevant solutions. In the scalar case, the paper analyzes monotonicity and TVNI properties of finite-difference schemes. It shows that monotone schemes are TVNI and that TVNI schemes are monotonicity preserving. The paper then describes how to convert first-order accurate TVNI schemes into second-order accurate TVNI schemes by modifying the flux function and introducing a numerical viscosity term. For systems of conservation laws, the paper extends the scalar scheme by applying it to each characteristic field. The resulting schemes are TVNI and second-order accurate. The paper also discusses the role of the function Q(x) in determining the numerical viscosity and the resolution of shocks and contact discontinuities. The paper concludes with a discussion of entropy enforcement and resolution in both genuinely nonlinear and linearly degenerate fields. It shows that the proposed schemes achieve high resolution and strong entropy enforcement, even in the presence of contact discontinuities. The paper also addresses the issue of numerical viscosity and its impact on the resolution of shocks and contact discontinuities.