A linear, second-order, unconditionally energy-stable scalar-auxiliary-variable scheme for the CHQ viscoelastic phase-separation model

Abstract

The CHQ model describes viscoelastic phase separation in a polymer--solvent mixture: a Cahn--Hilliard equation for the polymer volume fraction $ϕ$ is two-way coupled, through a phase-dependent cross-diffusion flux, to a damped relaxation equation for a scalar bulk elastic stress $q$. We propose a scalar-auxiliary-variable (SAV) time discretisation of this system, combining the SAV reformulation of the double-well potential with a second-order backward differentiation formula (BDF2) and a second-order extrapolation of all phase-dependent mobilities: a linear, computationally cheaper complement to the reference scheme's fully nonlinear Newton solve. The resulting method is (i) linear at every time step, with a coupling structure we prove has exactly one solution at every step, for every choice of time-step size; (ii) mass conserving, exactly, as an algebraic identity of the discrete equations themselves (the only drift ever seen numerically traces to the finite tolerance of the iterative linear solver, not a failure of the conservation property); and (iii) unconditionally stable for a modified discrete energy, via an exact one-step energy--dissipation identity. We show the energy identity extends to this filtered variant as well, with two explicit extra non-negative terms. Hence, the scheme actually producing the physics results is itself provably stable for the same energy. A Fourier--Galerkin implementation verifies that the energy identity holds at every time step to within floating-point round-off. It also confirms second-order temporal accuracy, including in the steep-interface regime, and agreement with an independently implemented nonlinear Newton--BDF2 solver. Compared with that independent nonlinear Newton--BDF2 solver, at equal grid resolution, the SAV method uses $2.5$--$3.0\times$ fewer linear-solver iterations and $1.7$--$2.0\times$ less wall-clock time.