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For several decades the capillary discharges have been under intensive investigations due to various promising applications, e.g. for the laser electron accelerators as well as for the X-ray lasers [1,2]. A major portion of the experiments were done with circular cross-section capillaries. An appropriate theoretical and numerical study of circular capillaries can be greatly simplified to a 1D model [3] assuming rotational and axial symmetries of the plasma flow in a long thin channel. On the other hand, studying capillaries with non-circular cross-section [4], which have been attracting substantially less attention, requires more complicated 2D models. Such capillaries, for example, square one, possess several advantages related to their fabrication as well as for plasma diagnostics The aim of our work is to compare the plasma density and temperature distributions formed at the quasistationary stage of the discharge. We present the results of MHD simulations of hydrogen-filled capillary discharges with circular and rectangular cross-sections under almost the same conditions characterizing the initial configurations and the external electric circuit. The simulation parameters are choosen to correspond to the capillary discharge based waveguide for the laser wakefield accelerator [5]. Bibliography [1] Leemans W. P. et al 2014 Phys. Rev. Lett. 113 245002 [2] Benware B. R. et al 1998 Phys. Rev. Lett. 81 5804 [3] Bobrova N. A. et al 2001 Phys. Rev. E 65 016407 [4] Gonsalves A. J. et al 2007 Phys. Rev. Lett. 98 025002 [5] Esarey E. et al 2009 Rev. Mod. Phys. 81 1229