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Implicit-explicit all-speed schemes for compressible Cahn-Hilliard-Navier-Stokes equations

Andreu Martorell, Pep Mulet, Dionisio F. Yáñez

Abstract

We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations in the low Mach number regime. The method is based on finite differences on staggered grids and is specifically designed to handle the challenges posed by the low Mach number limit, where the system approaches to an incompressible behavior. In this regime, standard explicit schemes suffer from severe time-step restrictions due to fourth-order diffusion terms and the stiffness induced by fast acoustic waves. To overcome this, we employ an IMEX strategy which splits the governing equations into stiff and non-stiff components. The stiff terms, arising from pressure, viscous forces and fourth-order Cahn-Hilliard contributions, are treated implicitly, while the remaining are dealt explicitly.

Implicit-explicit all-speed schemes for compressible Cahn-Hilliard-Navier-Stokes equations

Abstract

We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations in the low Mach number regime. The method is based on finite differences on staggered grids and is specifically designed to handle the challenges posed by the low Mach number limit, where the system approaches to an incompressible behavior. In this regime, standard explicit schemes suffer from severe time-step restrictions due to fourth-order diffusion terms and the stiffness induced by fast acoustic waves. To overcome this, we employ an IMEX strategy which splits the governing equations into stiff and non-stiff components. The stiff terms, arising from pressure, viscous forces and fourth-order Cahn-Hilliard contributions, are treated implicitly, while the remaining are dealt explicitly.
Paper Structure (24 sections, 2 theorems, 108 equations, 13 figures, 2 tables)

This paper contains 24 sections, 2 theorems, 108 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Cahn-Hilliard-Navier-Stokes Equations
  3. Model Description
  4. Isentropic compressible Cahn-Hilliard-Navier-Stokes in a low Mach number regime
  5. Numerical Schemes
  6. Spatial Semidiscretization
  7. Basic Finite Difference Operators
  8. The Operators $\mathcal{C}_1$ and $\mathcal{C}_2$
  9. The Operator $\mathcal{L}_1$
  10. The Operator $\mathcal{L}_2$
  11. The Operator $\mathcal{L}_3$
  12. The Operator $\mathcal{L}_4$
  13. Vector Implementation
  14. Implicit-Explicit Schemes
  15. Solution to the Nonlinear Systems
  16. ...and 9 more sections

Key Result

proposition thmcounterproposition

If $\varrho_k>0$ for every $k=1,\ldots,M^2$, and $\nu,\lambda>0$, then $D_\varrho + \Delta t\alpha_{i,i}B$ is symmetric and strictly positive definite.

Figures (13)

  • Figure 1: Diagram illustrates the asymptotic-preserving (AP) property. $\mathcal{M}^\delta$, $\mathcal{M}^0$ denotes the continuous compressible and incompressible system, while $\mathcal{M}_\Delta^\delta$, $\mathcal{M}_\Delta^0$ represents their discrete counterparts, respectively. The AP is verified if the diagram commutes.
  • Figure 2: Results for Test 1, $T=0, 0.01$, $M=128$ and $C_p=10^8$. Initially, $c$ is lies within the unstable region. At the beginning of the simulation, phase separation occurs. Moreover, the density starts to become higher in the lower part of the domain due to gravity.
  • Figure 3: Results for Test 1, $T=0.03, 0.05$, $M=128$ and $C_p=10^8$. The process of spinodal decomposition continues, and the density is accumulating at the bottom of the domain due to gravity.
  • Figure 4: Results for Test 1, $T=0.07, 0.1$, $M=128$ and $C_p=10^8$. It can be observed that density remains almost constant among distinct times and that phase separation has almost finished.
  • Figure 5: Results for Test 2, $T=0, 0.01$, $M=128$ and $C_p=10^8$. Initially, $c$ lies outside the unstable region and the density is dispersed. At $T=0.01$, the fluid starts to have denser regions near the bottom.
  • ...and 8 more figures

Theorems & Definitions (4)

  • proposition thmcounterproposition
  • definition thmcounterdefinition
  • theorem 1
  • proof