During this presentation, I explain mean curvature motion and the search for a higher-order method for solving the PDE. Algorithms for simulating mean curvature motion are only first-order accurate in the two-phase (or two grains) setting and their accuracy degrades further to half order in the multi-phase setting. I give a second-order two-phase method using threshold dynamics. The method loses energy stability, a desirable property of the original threshold dynamics algorithm. I then diverge to explain what does it take for a multi-stage (Runge-Kutta) numerical method for differential equations to be energy stable. I give new classes of energy stable, high order accurate Runge-Kutta schemes for gradient flows in a very general setting. I then finish with giving an energy stable, second-order accurate threshold dynamics algorithm for two dimensions. Speaker(s): Alexander Zaitzeff (Two Six Labs)