Convergence of a Decoupled Splitting Scheme for the Cahn–Hilliard–Navier–Stokes System

Chen Liu, Rami Masri, Béatrice Rivière

2023SIAM Journal on Numerical Analysis20 citationsDOI

Abstract

.This paper is devoted to the analysis of an energy-stable discontinuous Galerkin algorithm for solving the Cahn–Hilliard–Navier–Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable and mass conservative. The energy dissipation and the \(L^\infty\) stability of the order parameter are obtained under a CFL-like constraint. Optimal a priori error estimates in the broken gradient norm and in the \(L^2\) norm are derived. The stability proofs and error analysis are based on induction arguments and do not require any regularization of the potential function.KeywordsCahn–Hilliard–Navier–Stokesdiscontinuous Galerkinstabilityoptimal error boundsMSC codes65M1265M1565M60

Topics & Concepts

MathematicsNorm (philosophy)Cahn–Hilliard equationDissipationA priori and a posterioriDiscontinuous Galerkin methodRegularization (linguistics)Applied mathematicsMathematical proofStability (learning theory)Convergence (economics)Galerkin methodMathematical analysisFinite element methodPartial differential equationGeometryEconomicsMachine learningComputer scienceThermodynamicsLawPhysicsPhilosophyArtificial intelligenceEconomic growthEpistemologyPolitical scienceSolidification and crystal growth phenomenaAdvanced Numerical Methods in Computational MathematicsAdvanced Mathematical Modeling in Engineering