Hey all,

I'm looking for information about the performance of various numerical methods (i.e., forward euler, backward euler, RK methods, BDFs, etc.) with smoothed particle spatial methods. Seems like most SPH people only talk about explicit methods, and I was wondering why that is.

Thanks.

Dave

At first: note that SPH is a meshless method, but there are several alternative meshless methods that don't have your mentioned limitation (or attention), e.g. see seris of paper by Onate and Idelsohn (meshless finite element), or koshizoka and oka (moving particle semi-implicit, MPS) in spite of short life time, now there are several published book in this area!!!

Regarding to SPH: when it is proposed by monaghan for simulation of incompressible free surface flow, he try to make it commpetetive with mesh based methods (as it don't have mesh connectivity need search also generally its support set is wider than mesh based methods to acheive same accuracy) so take time step size small but don't attend to enforcing incompressibility, so original SPH was realised from solution of elliptic pressure equation and convert to fully explicit incompressible free surface flow solver. The others follow Monaghan (with open or closed eye). In my experience it is possible to run simulation with weakly incompressible Monaghan's SPH with CFL 0.1 (0.5 is for mesh based usually) so SPH has compettative behavior. Note that in theory stable CFL must be calculated based on sound speed, for more see monaghan article.

Also some people developed incompressible SPH (same as other meshless methods) with solution of pressure poisson equation for enforcing incompressibility (so your review seems incomplete), for this see paper by Cummins and Rudmany (in JCP, 1999) or by Shao and Lo (Advances in Water Resources, 2003). The main drawback of these approach is very high computational cost while the accuracy is not so good (i personally test it also have several private communication with developers that confirm my conculation). So one reason for this fact that why all people are biased to explicit SPH can be this (inefficiency of implicit ones).

Hope this help.