Reformulation of Smoothed Particle Hydrodynamics with Riemann Solver

Smoothed Particle Hydrodynamics is reformulated in terms of the convolution of the original hydrodynamics equations, and the new evolution equations for the particles are derived. The same evolution equation of motion is also derived using a new action principle. The force acting on each particle is determined by solving the Riemann problem. The use of the Riemann Solver strengthens the method, making it accurate for the study of phenomena with strong shocks. The prescription for the variable smoothing length is shown. These techniques are implemented in strict conservation form. The results of a few test problems are also shown.