We study reconnections of quantum vortices by numerically solving the governing Gross-Pitaevskii equation. We find that the minimum distance between vortices scales differently with time before and after the vortex reconnection. We also compute vortex reconnections using the Biot-Savart law for vortex filaments of infinitesimal thickness, and find that, in this model, reconnection are time-symmetric. We argue that the likely cause of the difference between the Gross-Pitaevskii model and the Biot-Savart model is the intense rarefaction wave which is radiated away from a Gross-Pitaeveskii reconnection. Finally we compare our results to experimental observations in superfluid helium, and discuss the different length scales probed by the two models and by experiments.

We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O((log n) 2 ). To the best of our knowledge this is the first complete, explicit example of near optimal bivariate interpolation points.

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We propose and analyze the ReLPM (Real Leja Points Method) for evaluating the propagator '(tB)v via matrix interpolation polynomials at spectral Leja sequences. Here B is the large, sparse, nonsymmetric matrix arising from stable 2D or 3D nite-dierence discretization of linear advection-diusion equations, and '(z) is the entire function '(z) = (e z − 1)=z. The corresponding sti dierential systemẏ(t) = By(t) + g; y(0) = y 0 , is solved by the exact time marching scheme y i+1 = y i + t i '(t i B)(By i + g), i = 0; 1; : : : ; where the time-step is controlled simply via the variation percentage of the solution, and can be large. Numerical tests show substantial speed-ups (up to one order of magnitude) with respect to a classical variable step-size Crank-Nicolson solver.

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