Integrated gradients: A derivation of some difference forms for the equation of motion for compressible flow in two-dimensional Lagrangian hydrodynamics, using integration of pressures over surfaces

This paper describes a method of deriving gradients (that is, accelerations) for difference calculations of the equations of motion (momentum conservation) in two-dimensional Lagrangian meshes in an r-z coordinate system. The method basically considers various ways of defining the masses associated with each vertex and methods of integrating pressures over the surfaces of those masses, and then combining them in various ways to conserve momentum transfer between vertices. These gradients are derived analytically for planes, cylinders, and spheres to test for uniform motion. All results described have been tested with actual numerical calculations.