Vladimir Molchanov

• Recently, the Hamiltonian Particle-Mesh (HPM) method was proposed by J.Frank, G.Gottwald and S.Reich. It incorporates the procedure of artificial smoothing over a regular grid into the classical Smoothed Particle Hydrodynamics (SPH) formalism to improve the long-time stability of the method. HPM was applied to numerical solution of Shallow Water Equations (SWE), which are extensively used to simulate geophysical processes. So far, little research has been done to analyze and optimize HPM. The primary focus of our work is on the convergence of HPM method to solutions of a class of barotropic fluid equations. The convergence is investigated as smoothing scales vanish and as the number of particles grows unboundedly. The aim is to find an explicit relationship between these parameters to provide the best rate of convergence. For that purpose we adapt the convergence result for SPH obtained by K. Oelschläger. We consider HPM applied to the fluid model including SWE as a partial case and generalize the result to a self-gravitating fluid. Two new high-order meshfree methods for two-dimensional rotating shallow flows are proposed. They combine the conservation of Ertel potential vorticity with adaptive quadrature weights or moving least square approach for interpolation of scattered data. Generalizations for three-dimensional flows are discussed.