This work concerns the development of a meshfree semi-implicit numerical scheme based on the Smoothed Particle Hydrodynamics (SPH) method, here applied to free surface hydrodynamic problems governed by the shallow water equations. In explicit numerical methods, a severe limitation on the time step is often due to stability restrictions imposed by the CFL condition. In contrast to this, we propose a semi-implicit SPH scheme, which leads to an unconditionally stable method. To this end, the discrete momentum equation is substituted into the discrete continuity equation to obtain a linear system of equations for only one scalar unknown, the free surface elevation. The resulting system is not only sparse but moreover symmetric positive definite. We solve this linear system by a matrix-free conjugate gradient method. Once the new free surface location is known, the velocity can directly be computed at the next time step and, moreover, the particle positions can subsequently be updated. The resulting meshfree semi-implicit SPH method is validated by using a standard model problem for the shallow water equations.
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