I am trying to understand the meshless methods as described in https://arxiv.org/pdf/1409.7395.pdf. I'm having trouble understanding the following step: (Page 7, just after equation 17)

Now, rather than take the flux functions F directly at the particle location and time of i or j, in which case the scheme would require some ad-hoc artificial dissipation terms (viscosity and con- ductivity) to be stable, we can replace the fluxes with the solution of an appropriate time-centered Riemann problem between the particles/cells i and j, which automatically includes the dissipation terms.

What I can't find informations about is: What is a time-centered Riemann problem? I wasn't able to find any details on this so far, not even in Toro's "Riemann Solvers and Numerical Methods for Fluid Dynamics". From what I know, a Riemann problem is a special initial value problem for a conservation law

$$ \mathbf{U}_t + \mathbf{A}(\mathbf{U}) \mathbf{U}_x = 0$$

with

$$ \mathbf{U}(x, 0) = \mathbf{U}_0(x) = \cases{ \mathbf{U}_L \text{ if } x<0\\ \mathbf{U}_R \text{ if } x>0} $$

So how does time-centering come into play here?

  • It's not a standard term; the authors are just trying to describe their algorithm. – David Ketcheson Dec 6 at 20:16

Some schemes have a two step update. First update solution $u^n$ to middle of time interval $[t_n,t_{n+1}]$ to get $u^{n+1/2}$. Then use this predicted solution $u^{n+1/2}$ to do the final update, where you make use of Riemann solvers. This is what they probably mean by time centered Riemann solver. In particular, search for MUSCL-Hancock scheme for more details on this approach.

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