Convergence of Numerical Schemes for a Conservation Equation with Convection and Degenerate Diffusion (Journal of Computational Mathematics, 2021)
The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media withthermodynamic transitions, is done using a $θ$-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the testfunctions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result forthe time translate in the gradient discretisation setting. Some numerical examples, usingboth the Control Volume Finite Element method and the Vertex Approximate Gradientscheme, show the role of $θ$ for stabilising the scheme.
Keywords: Linear convection; Degenerate diffusion; Gradient discretisation method; $θ$-scheme.