In this paper, inspired by the second-order dynamic system with Hessian-driven damping, we combine the implicit and explicit discretization of the velocity term \(\dot{x}(t)\) in differential equations and propose an improved accelerated gradient method for smooth convex optimization problem, which has both stability and good numerical performance. We present the convergence result of iteration points and obtain convergence rate of \(o(1/k^{2})\) for the function values. Similarly, we propose an improved proximal gradient method with extrapolation to solve composite convex optimization problem, which also has \(o(1/k^{2})\) convergence rate for the function values. We conduct some numerical experiments on the differentiable convex minimization problem, the lasso problem, and the \(\ell _{1}\) regularized logistic regression to demonstrate the advantage of the proposed methods.
