This, BTW, is why we developed a semi-implicit formulation that handles the viscous part (only) implicitly:
https://www.sciencedirect.com/science/article/abs/pii/S002199911830593X
https://www.sciencedirect.com/science/article/abs/pii/S0021999122004752
(the latter may be of interest to you if you use #BiCGSTAB or #ConjugateGradient methods to solve linear systems that have nothing to do with SPH, BTW).

And now comes the new difficulty: THERMAL!

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