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A new high‐order weighted essentially non‐oscillatory scheme for non‐linear degenerate parabolic equations
Author(s) -
Abedian Rooholah
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22581
Subject(s) - stencil , mathematics , dimension (graph theory) , degenerate energy levels , gravitational singularity , scheme (mathematics) , order (exchange) , nonlinear system , mathematical analysis , pure mathematics , physics , computational science , finance , quantum mechanics , economics
Abstract In this research, a new high‐order WENO method for solving 1D, 2D and 3D non‐linear degenerate parabolic equations which may contain discontinuous solutions has been designed. The new scheme is constructed by three polynomials of varying degrees. This new method applies the same six‐point stencil as the classic WENO procedures developed for degenerate parabolic by Liu et al. and Abedian et al. (Comput. Phys. Commun., 184 (2013), 1874–1888). The new scheme attains sixth‐order accuracy in smooth areas and second‐order accuracy near singularities. Comparing the magnitude of errors produced by Liu et al., Abedian et al. and the new scheme has shown that the new scheme performs better. The only condition for the linear weights (ideal weights) associated with this scheme is that their sum must be equal to one, while they can be any positive real number. Afterwards, the non‐oscillatory weights are calculated with the linear weights. The new proposed scheme can be easily developed to higher dimension by dimension‐by‐dimension technique. The new scheme enjoys simplicity, good performance and low computational cost. To explain the properties of the method such as accuracy and resolution, a number of classical numerical examples have been prepared.