Xuelong Gu

Xuelong Gu, Qi Wang

The skew-gradient embedding (SGE) framework~\cite{GuWangSGE2025} reformulates a thermodynamically consistent system as a generalized gradient flow by embedding its zero-energy contribution in a skew-symmetric operator. In a time-discrete scheme, the profiles defining this operator may be evaluated at previous time levels. The resulting operator remains skew-symmetric, so its contribution to the discrete energy balance vanishes; this explicit treatment often decouples multiphysics systems. We show that this operator is not unique: the admissible gauges form an affine space, and we call the resulting family generalized skew-gradient embeddings (GSGE). For any positive definite metric, least squares selects a unique minimum-Hilbert--Schmidt gauge, and the native metric recovers SGE. This construction also gives regularized approximations, corrections of non-neutral residuals, and gauges that preserve prescribed invariants. For rank-two gauges, we use a necessary and sufficient Jacobi criterion. Applying this criterion to a compatible MAC discretization of the incompressible Navier--Stokes equations gives a finite-dimensional rank-two Poisson--GENERIC formulation at the semi-discrete level; the implicit midpoint rule preserves this rank-two GENERIC structure at the fully discrete level and satisfies the exact discrete energy law. For the Cahn--Hilliard--Navier--Stokes system, the regularized GSGE--BDF2 scheme preserves mass, dissipates the discrete energy unconditionally, and admits a decoupled implementation.

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.