Accurate particle splitting for smoothed particle hydrodynamics in shallow water with shock capturing

The solution for the shallow water equations using smoothed particle hydrodynamics is attractive, being a mesh-free, automatically adaptive method without special treatment for wet–dry interfaces. However, the relatively new method is limited by the variable kernel size or smoothing length being inversely proportional to water depth causing poor resolution at small depths. Boundary conditions at solid walls have also not been well resolved. To solve the resolution problem in small depths, a particle splitting procedure was developed (conveniently into seven particles), which conserves mass and momentum by varying the smoothing length, velocity and acceleration of each refined particle. This improves predictions in the shallowest depths where the error associated with splitting is reduced by one order of magnitude in comparison to other published works. To provide good shock capturing behaviour, particle interactions are treated as a Riemann problem with Monotone Upstream-centred Scheme for Conservation Laws (MUSCL) reconstruction providing stability. For solid boundaries, the recent modified virtual boundary particle method was developed further to enable the zeroth moment to be accurately conserved where the smoothing length of particles is changing rapidly during particle splitting. The resulting method is applied to the one-dimensional and the two-dimensional axisymmetric wet-bed dam break problems showing close agreement with analytical solutions, demonstrating the need for particle splitting. To demonstrate wetting and drying in a more complex case, the scheme is applied to oscillating water in a two-dimensional parabolic basin and produces good agreement with the analytical solution. The method is finally applied to the European Concerted Action on DAm break Modelling dam-break test case representative of realistic conditions and good predictions are made of experimental measurements with a 40% reduction in the computational time when particle splitting is employed. The overall method has thus become quite sophisticated but its generality and versatility will be attractive for various shallow water problems