Xuelong Gu

Xuelong Gu, Qi Wang

We develop an energy-stable implicit convex-splitting BDF2 discretization (CS-BDF2) of the Cahn--Hilliard--Navier--Stokes equations. For the Cahn--Hilliard equation, BDF2 analyses can establish energy stability by testing the phase equation in the (H^{-1}) metric. For CHNS, this test is not compatible with the coupled energy estimate: the momentum equation is tested by (\bfu^{n+1}), while the transported phase equation is tested by (μ^{n+1}) so that transport cancels capillary work. The chemical-potential relation must then be paired with the BDF2 phase increment ((3ϕ^{n+1}-4ϕ^n+ϕ^{n-1})/2); its nonlinear part must produce a BDF2 bulk-energy difference, up to nonnegative higher-order history terms. To overcome this difficulty, we introduce a new BDF2-compatible convex-splitting approximation of the nonlinear bulk force that directly yields a discrete bulk-energy identity and enables a discrete energy analysis for the CHNS system. Specifically, we discretize the bulk force (f(ϕ)=ϕ^3-ϕ) by (χ(ϕ^{\dagger,n+1},ϕ^{\dagger,n})-ϕ^{*,n+1}), where (χ(a,b)=\tfrac14(a^2+b^2)(a+b)), (ϕ^{\dagger,n+1}=\tfrac{3ϕ^{n+1}-ϕ^n}{2}), (ϕ^{\dagger,n}=\tfrac{3ϕ^n-ϕ^{n-1}}{2}), and (ϕ^{*,n+1}=2ϕ^n-ϕ^{n-1}). This discretization is based on the shifted BDF2 identity ((3ϕ^{n+1}-4ϕ^n+ϕ^{n-1})/2=ϕ^{\dagger,n+1}-ϕ^{\dagger,n}). With a matching discretization of the reversible coupling terms in CHNS, the scheme is mass conservative, uniquely solvable, and unconditionally energy stable. We prove second-order convergence for the phase variable, chemical potential, velocity, and pressure.

Xuelong Gu, Qi Wang

We present a nonlinear multigrid solver for the Richards equation in variably saturated porous media with strongly nonlinear hydraulic conductivity and water-retention relationships. The governing equation is discretized using a second-order conservative finite-difference scheme in space and an implicit backward differentiation formula in time. The core component of the solver is a nonlinear Gauss--Seidel (NGS) smoother based on a triangular splitting of the diffusion operator combined with diagonal stabilization. This construction yields a sequence of locally decoupled scalar nonlinear problems that can be solved efficiently and robustly using only a few Newton iterations. Under suitable monotonicity assumptions, we establish the convergence of the NGS iteration in the $L^\infty$ norm and derive explicit conditions on the stabilization parameters. Numerical experiments for benchmark infiltration, drainage, and root-uptake problems demonstrate that the proposed NGS-based multigrid framework is both computationally efficient and robust.

Xuelong Gu, Qi Wang

We develop a structure-preserving computational framework for acoustic wave scattering by moving objects, comprising a new PML-domain-embedding model and a compatible numerical approximation. The model couples a perfectly matched layer (PML), used to truncate the acoustic wave equation, with a domain-embedding formulation that represents moving objects on a fixed computational domain. The resulting PML-domain-embedding (PML-DE) system enables moving-boundary scattering problems to be solved without remeshing. Using matched asymptotic expansions, we show that the diffuse-interface formulation converges to the corresponding sharpinterface system as the interface thickness tends to zero. We then construct an energy-dissipationrate-preserving finite-difference scheme for the PML-DE system. To improve computational efficiency, the scheme is combined with hierarchical local refinement informed by the moving-object location, the fixed PML region, and the evolving wave dynamics, all within the fixed computational domain. Numerical experiments demonstrate the accuracy of the computed scattering solutions, the effectiveness of the absorbing layer and object-embedding strategy, and the efficiency of the adaptive algorithm. The proposed framework provides a practical and robust computational approach for engineering applications involving complex acoustic wave-scattering problems.

Xuelong Gu, Guanghua Ji, Qi Wang

We propose several linear, fully decoupled numerical schemes with first- and second-order temporal accuracy for a novel Q-tensor-based two-phase hydrodynamic model describing the coupling of active nematic liquid crystal solutions with isotropic solid substrates. The model is derived from the generalized Onsager principle and includes nontrivial terms that contribute zero to the total free-energy dissipation. We prove that the proposed decoupled linear schemes are thermodynamically consistent at the discrete level. In the passive limit, the SGE-BDF1 and SGE-PDG schemes are unconditionally energy stable, while the SGE-BDF2 scheme is energy stable with respect to a modified energy under a standard boundedness assumption and a sufficiently large stabilization parameter. We perform extensive numerical simulations to investigate how activity and other model parameters affect active nematic fluid-solid interactions. Finally, we analyze the physical mechanisms underlying the observed behaviors, providing deeper insight into the dynamics of soft confined active nematic fluids.