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tokens: concept beyond one dimension (1D) or to unstructured/meshfree methods is less clear (e.g. Chiapolino, Saurel Nkonga 2017 ). A physical interpretation can be seen from our equation ( 6 ). For the drag term to provide a positive definite contribution to the entropy, and must have the same sign, such that d u /d t | drag is positive. Pairwise positivity is not strictly necessary so long as the sum over all neighbours is positive. We tried setting if the signs differ, but found this to be more dissipative than using slope limiters (see Fig. 3 ). We found the van Leer MC limiter to provide the best compromise between monotonicity and dissipation. 3 RESULTS We test our improved algorithm in 1D using the ndspmhd code (Price 2012 ) and in 3D using phantom (Price et al. 2018 ). We use explicit global time-stepping with a leapfrog integrator, the M 6 quintic kernel for the SPH terms with the double hump M 6 employed for the drag terms ( LP12a ). The results are not sensitive to the choice of kernel, provided a double hump kernel is used for the drag. The time-step was set to 0.9 times the minimum stopping time (we found that setting Δ t = t s exactly as in LP12a could result in instability with reconstruction). We use the van Leer MC limiter unless otherwise specified. 3.1 DUSTYWAVE Fig. 1 shows the results of the dustywave described in Laibe Price ( 2011 ), performed using 2 × n x particles with a fixed drag coefficient K = 1000, ρ g = ρ d = 1 and c s = 1 (giving t s = 5 × 10 −4 ) and a perturbation amplitude of 10 −6 . We use an adiabatic equation of state P = ( γ − 1) ρu with γ = 5/3 in the gas. In the absence of reconstruction, overdamping occurs when h ≳ c s t s , i.e. for n x ≲ 1024 (left-hand column), as found by LP12a . Adding reconstruction captures the true solution to within a few per cent for n x ≳ 64 (middle column), while the slope limiter does not visibly degrade it (right-hand column). Figure 1. Dust and gas velocities in the dustywave test after 10 wave periods, using K = 1000 with 2 × n x particles without reconstruction and with and without the slope limiter (see labels). Reconstruction avoids the need to resolve h ∼ t s c s (resolved at n x = 1024 particles). Exact solution shown in red. Fig. 2 shows the results in 3D using phantom . We follow the procedure used in Price et al. ( 2018 ), placing the particles using dense sphere packing and cropping the grid in the y and z directions at 12 particle spacings (for efficiency), giving 2 × 128 × 12 × 12 particles. The results in 3D are indistinguishable from those shown in Fig. 1 , showing our method also works in 3D. Figure 2. As in Fig. 1 , but in 3D with phantom using n x × 12 × 12 gas particles (solid) and n x × 12 × 12 dust particles (open) initially placed using dense sphere packing. Exact solution from Laibe Price ( 2011 ) shown in red. 3.1.1 Choice of slope limiter Fig. 3 shows the kinetic energy as a function of time in the 1D dustywave problem at a resolution of n x = 128. The