An upwind DG scheme preserving the maximum principle for the convective Cahn-Hilliard model

Abstract

The design of numerical approximations of the Cahn-Hilliard model preserving the maximum principle is a challenging problem, even more if considering additional transport terms. In this work, we present a new upwind discontinuous Galerkin scheme for the convective Cahn-Hilliard model with degenerate mobility which preserves the maximum principle and prevents non-physical spurious oscillations. Furthermore, we show some numerical experiments in agreement with the previous theoretical results. Finally, numerical comparisons with other schemes found in the literature are also carried out.

Energy-stable and boundedness preserving numerical schemes for the Cahn-Hilliard equation with degenerate mobility

“…Mixed finite element approximations using logarithmic potentials and degenerate mobilities were studied in [3,7]. More recently, several discontinuous Galerkin methods have been considered for the problem with and without convection [1,16,17,25].…”

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

“…Applying the maximum principle, they investigated the generalized time-fractional variable-order Caputo diffusion equations and fractional variable-order Riesz-Caputo diffusion equations. For other new developments of the maximum principle, the reader can refer to [8][9][10][11][12][13][14][15][16] and the references therein.…”

An Unconditionally Energy Stable and Positive Upwind DG Scheme for the Keller–Segel Model

“…In the last decades, a lot of papers have been published dealing with these kinds of theoretical issues both for the classical model (1) and for other models based on some extensions or generalizations. In general, they start from the Keller-Segel classical equations and modify them with the purpose of avoiding the non-physical blow up of solutions, producing solutions which are closer to the "real chemotaxis" phenomena observed in biology.…”

“…Despite that, many interesting works have been published on numerical simulation of chemotaxis equations using different kinds of approaches. For instance, the work by Saito [35] uses FE with upwind stabilization for the parabolic-elliptic Keller-Segel model [τ = 0 in (1)] showing mass conservation, positivity and error estimates. Also, Gutiérrez-Santacreu and Rodríguez-Galván demonstrate positivity, an energy law and a priori bounds for their FE scheme on acute meshes in [25].…”

“…In particular, in Sect. 4.1, we define the upwind approximation based on the ideas introduced in [1] along with some geometrical considerations for the mesh family T h , and we discuss the properties of the scheme in Sect. 4.2.…”