An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions

The smoothed particle hydrodynamics (SPH) method has proven useful for modeling large deformation of fluids including fluids with stress‐free surfaces. Because of the Lagrangian nature of the method, it is well suited to address the thermal evolution of these free surface flows. Boundary conditions at the interface of the fluid with a solid wall are usually enforced through the use of boundary particles. However, applying conditions at free surfaces, in particular gradient boundary conditions, can be problematic with traditional SPH formulations due to the degradation of the gradient approximation in these regions. Compounding this difficulty is that traditional approximations of the Laplacian operator suffer a similar degradation near free surfaces. A new SPH formulation of the Laplacian operator is presented, which improves the accuracy near free surface boundaries. This new form is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems, but includes higher‐order terms in the appropriate Taylor series. Comparisons with other approximations of second‐order derivatives are given. The discretization is tested by solving steady‐state and transient problems of thermal diffusion using the Backward Euler method with a GMRES solver. Boundary conditions are imposed through an augmented matrix. Copyright © 2008 John Wiley & Sons, Ltd.