Sondre Tesdal Galtung, NTNU – Norwegian University of Science and Technology
In this talk we will consider a discretization of the Camassa–Holm equation based on variational principles in Lagrangian coordinates, which has been shown to converge to so-called conservative solutions. These are solutions which satisfy an additional balance law for the energy density of the equation, ensuring that the total energy is conserved globally in time. The corresponding numerical method in a periodic domain performs well for several traveling-wave reference solutions typical for the CH literature, e.g., the well-known peakons, and even for reference solutions involving wave-breaking and energy concentration. However, when “challenged” to apply our method to less prevalent traveling waves, known as stumpons, we were led to some unexpected results. Based on this observation, we prove that stumpons are non-conservative and hence not suitable for approximation with our numerical method.