The first order symmetric SPH method for transient heat conduction problems

In order to improve the accuracy and the stability of the conventional smoothed particle hydrodynamics (SPH) method for simulating the transient heat conduction problems, a first order symmetric smoothed particle hydrodynamics (FO-SPH) method is proposed. In order to solve the heat conduction problem with second derivative, the proposed FO-SSPH method is first to decompose the problem into two first order partial differential equations (PDEs), and then the first order kernel gradient is corrected based on the discretization of gradient and the concept of Taylor series. Finally, the obtained local matrix is locally symmetrized. All the numerical results demonstrate that the FO-SSPH possesses a higher accuracy and better stability than the SPH method, that the mixed boundary conditions can be well imposed using FO-SSPH method, and that the reliability and the flexibility of the FO-SSPH method can also be observed for PDEs with multi-boundary conditions. Finally, the one-dimensional nonlinear heat conduction problem is investigated by the FO-SSPH method, and the phenomena of concave and bulge are observed when the temperature achieves the stable state, in which the influence of the coefficients for heat flux is discussed.