Runge-Kutta vs Taylor-SPH: Two time integration schemes for SPH with application to Soil Dynamics Articles uri icon



publication date

  • March 2013

start page

  • 3541

end page

  • 3563


  • 5


  • 37

International Standard Serial Number (ISSN)

  • 0307-904X

Electronic International Standard Serial Number (EISSN)

  • 1872-8480


  • A comparison between two different time integration schemes for SPH applied to Soil Dynamics problems is presented. The traditional SPH method consists of applying first the SPH spatial discretization to the governing equations, obtaining a set of simultaneous ordinary differential equations with respect to time. This set of equations is then integrated in time using one of the standard techniques such as one of the Runge–Kutta schemes. In most cases, governing equations are formulated in terms of displacements, and numerical integration of the field variables is carried out at every particle, in most cases using an Eulerian kernel, giving rise to numerical instabilities. It will be shown here that it is possible to minimize these numerical problems using the mixed stress-velocity formulation along with a corrected SPH for the spatial discretization. Nevertheless, in presence of discontinuous functions, such as shock waves, these methods present still some numerical difficulties. To solve the problem of shock waves propagation, the authors propose in this paper the use of an alternative method: the Taylor-SPH (TSPH), which consists of applying first the time discretization by means of a Taylor series expansion in two steps and thereafter the space discretization using a corrected SPH. The TSPH time integration algorithm is shown to be more robust, stable and efficient than the Runge–Kutta time integration schemes for SPH, and only a reduced number or particles is required to obtain accurate results.


  • taylor-sph (tsph); runge kutta; meshfree method; shock wave; viscoplastic; soil dynamics