en تخصصي Special پژوهشي Research paper <br> <div dir="ltr" style="text-align: right;"></div> <p style="margin: 0px; text-align: justify;">This paper provides a fourth-order scheme for approximating solutions of non-linear degenerate parabolic equations that their solutions may contain discontinuity. In the reconstruction step, a fourth-order weighted essentially non-oscillatory (WENO) reconstruction in Legendre basis, written as a convex combination of interpolants based on different stencils, is constructed. In the one-dimensional case, the new fourth-order reconstruction is based on a four-point stencil. The most important subject is that one of these interpolation polynomials is taken as a quadratic polynomial, and the linear weights of the symmetric and convex combination are set as to get fourth-order accuracy in smooth areas. Following the methodology of the traditional WENO-Z reconstruction, the non-oscillatory weights is calculated by the linear weights. The accuracy, robustness, and high-resolution properties of the new procedure&nbsp;<span style="text-align: justify;">are shown by extensive numerical examples.</span></p> WENO schemes, Legendre orthogonal polynomials, multidimensional non-linear degenerate parabolic equations, porous medium equation. 125 143 http://ijmsi.ir/browse.php?a_code=A-10-3361-1&slc_lang=en&sid=1 R. Abedian rabedian@ut.ac.ir 10031947532846008810 10031947532846008810 Yes University of Tehran