Development of time-space adaptive smoothed particle hydrodynamics method with Runge-Kutta Chebyshev scheme

Abstract Smoothed particle hydrodynamics (SPH) method is a Lagrangian particle method that has been widely used for solving complex mechanics problems. To reduce the SPH computational cost, the adaptive SPH (ASPH) with time-varying particle distribution has been proposed using the particle splitting and merging techniques. However, the particle splitting in the adaptive SPH results in a decreased time step due to the reduced particle spacing and thus requires an increased amount of computing time. In the present work, an explicit Runge-Kutta Chebyshev time stepping scheme is implemented within the adaptive SPH, by including the stabilizing substeps into the Runge-Kutta integration method with the use of the Chebyshev polynomials. The number of stabilizing substeps in a single SPH time step can be adaptive after particle splitting while the Runge-Kutta integration time step remains the same. Therefore, the proposed SPH algorithm is adaptive in both time and space domains. Representative example problems are simulated using the developed time-space adaptive SPH method. It is demonstrated that, as compared to the standard SPH and the conventional adaptive SPH with varying particle distribution only, the presented time-space ASPH method exhibits much higher efficiency without compromising accuracy and the superiority in terms of stability.