In this work, we develop an enhanced particle shifting strategy in the framework of weakly compressible ${\delta }^{+}$-SPH method. This technique can be considered as an extension of the so-called improved particle shifting technology (IPST) proposed by Wang et al. (2019). We introduce a new parameter named $“\varphi ”$ to the particle shifting formulation, on the one hand to reduce the effect of truncated kernel support on the formulation near the free surface region, on the other hand, to deal with the problem of poor estimation of free surface particles. We define a simple criterion based on the estimation of particle concentration to limit the error’s accumulation in time caused by the shifting in order to achieve a long time violent free surface flows simulation. We propose also an efficient and simple concept for free surface particles detection. A validation of accuracy, stability and consistency of the presented model was shown via several challenging benchmarks.

In this work we present a $\delta$-Smoothed Particle Hydrodynamics (SPH) scheme for weakly compressible flows with automatic adaptive numerical dissipation. The resulting scheme is a meshless self-adaptive method, in which the introduced artificial dissipation is designed to increase the dissipation in zones where the flow is under-resolved by the numerical scheme, and to decrease it where dissipation is not required. The accuracy and robustness of the proposed methodology is tested by solving several numerical examples. Using the proposed scheme, we are able to recover the theoretical decay of kinetic energy, even where the flow is under-resolved in very coarse particle discretizations. Moreover, compared with the original $\delta$-SPH scheme, the proposed method reduces the number of problem-dependent parameters.

The present paper proposes an iterative implicit numerical method for simulating the thaw depth of permafrost soil. For this purpose, the enthalpy-porosity model was used for the phase change process, and the finite difference scheme FTCS (Forward Time Centered Space) was used for discretization. An artificial mushy zone was maintained with the same thickness by keeping the regularization parameter proportional to the temperature gradient. In doing so, we made the scheme more stable and convergence occurred faster. The model accuracy was validated by comparing the numerical results with the analytical Stefan solution and with the results of a derived numerical model, based on an explicit scheme. The model performance was also tested against observation data collected on four different landscapes with different soil profiles and located on a basin underlain by continuous permafrost. It was found that the proposed model matched noticeably well the analytical solution for a volumetric liquid fraction (phi) equal to 0.5 regardless of the grid resolution. Furthermore, compared with the observation data, the model reproduced the annual maximum thaw depth with an absolute error lying between 0.7 and 7.7%. In addition, the designed algorithm allowed the model to converge after a maximum of eight iterations, reducing the computational time by around 75% compared to the explicit model. The results were so encouraging that the model can be included in a hydrological modeling of permafrost watersheds or cold regions in general.