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\title{A Spatially Staggered Particle Method Using the Strong Form}
\author{L. Libersky}

 \date{ }

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\maketitle

We will give an update of our work on the meshfree particle method Dual
Particle Dynamics (DPD) and a short review of SPH. The potential of
meshfree methods for problem solving is huge, but so are the problems
incurred by discarding the mesh. We will talk about difficulties that
have been overcome and those that remain. Specifically, we will address
questions of stability. It is shown that DPD and SPH can be made
conditionally stable for Eulerian kernels and linear fields. This result
is important because it is highly desirable to move and change neighbors
where the material deformation is large. For higher dimensions (than
1D), stability for general neighborhoods is shown to require a two-step
update, such a predictor-corrector scheme. The apparent tension between
completeness and conservation will also be addressed.

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