Hence, for 2D surface waves in weakly incompressible fluids, basic SPH formulation without any modification is as good as the R-SPH or LES-SPH methods. It is also observed that the use of Renormalized SPH (R-SPH) increases diffusivity but does not increase accuracy significantly. Furthermore, application of the LES filter has negligible effects. The method is meshfree in the sense that there are no elements or other geometrical connections between the nodes. If the artificial viscosity used is small, results indicate that the accuracy of SPH scheme is reasonably good however, a low artificial viscosity leads to a rugged air-water interface. Then, simulations of 2D free-surface waves in a weakly incompressible fluid are conducted. Application of SP approximations to Euler and Navier-Stokes equations is discussed, followed by an improvement to restore first order consistency in SPH. To simulate some problems, the partial differential equations and boundary conditions for two dimensional solid mechanics and fluid mechanics problem and heat. In this thesis, a basic introduction to particle and Smoothed Particle (SP) approximations is given first. Smoothed Particle Hydrodynamics (SPH) is a Lagrangian-based numerical method used for simulating problems in fluid and solid mechanics.
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