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In this paper, we study damped second-order dynamics, which are quasilinearhyperbolic partial differential equations (PDEs). This is inspired by therecent development of second-order damping systems for accelerating energydecay of gradient flows. We concentrate on two equations: one is a dampedsecond-order total variation flow, which is primarily motivated by theapplication of image denoising; the other is a damped second-order meancurvature flow for level sets of scalar functions, which is related to anon-convex variational model capable of correcting displacement errors in imagedata (e.g. dejittering). For the former equation, we prove the existence anduniqueness of the solution. For the latter, we draw a connection between theequation and some second-order geometric PDEs evolving the hypersurfaces whichare described by level sets of scalar functions, and show the existence anduniqueness of the solution for a regularized version of the equation. Thelatter is used in our algorithmic development. A general algorithm fornumerical discretization of the two nonlinear PDEs is proposed and analyzed.Its efficiency is demonstrated by various numerical examples, where simulationson the behavior of solutions of the new equations and comparisons withfirst-order flows are also documented.

A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science