Thomas Christiansen, NTNU – Norwegian University of Science and Technology
A wide class of solutions to the Hunter–Saxton equation is known to experience wave breaking in finite time. This phenomenon is characterized by the spatial derivative becoming pointwise unbounded from below, whilst the solution itself remains bounded and continuous. Moreover, energy concentrates on sets of zero measure. Consequently, if one wants to continue the solution past wave breaking, one must choose how to manipulate the concentrated energy. In this talk we consider a numerical method for the class of \(\alpha\)-dissipative solutions, corresponding to the case where we at each wave breaking occurrence remove an $\alpha$-fraction of the concentrated energy. The method is based on a novel projection operator and combines that with exact evolution along characteristics. We will motivate the construction of this projection operator and indicate how to show convergence. In addition, we briefly examine a condition guaranteeing a convergence rate.