Mobility Operator Splitting Technique

• Mobility operator splitting is a numerical technique that decouples mobility-driven operators from nonlinear terms, enhancing simulation efficiency and stability.
• It is applied in gradient flows, phase-field, and cross-diffusion models by separating reaction and diffusion steps to simplify complex PDEs.
• The method preserves energy stability and mass conservation while enabling robust variational and explicit schemes for state-dependent mobility functions.

The mobility operator splitting technique refers to a class of numerical and variational operator-splitting strategies in the simulation of evolution equations, where the mobility—often a scalar or matrix-valued function, possibly state- or density-dependent—determines the division of computational steps. This class of methods is critical in gradient flows, cross-diffusion systems, phase field models, structured population dynamics, and conservative-dissipative PDEs, where separating the mobility-dependent (often diffusive or transport) operator from nonlinear or nonlocal terms enables both analytical tractability and high-performance algorithms.

1. Theoretical Foundation

Mobility operator splitting arises primarily in gradient flows of the form

$$\partial_t u = \nabla \cdot \left(M(u) \nabla \frac{\delta \mathcal{F}}{\delta u} \right),$$

where $u$ (possibly vector-valued) denotes the state variable, $M(u)$ is the mobility (a symmetric positive (semi-)definite matrix function), and $\mathcal{F}$ is a free energy functional. The key rationale is to decouple the evolution into substeps that each deal with either the mobility-driven (linear or semi-linear) part or the nonlinear/nonlocal part of the dynamics.

This approach can be understood in both the time-discretized operator splitting—explicitly partitioning time integration into subtasks—and the variational minimizing-movement (JKO) setting, where gradient flows are recast as sequences of energy minimizations with transport costs induced by the mobility.

2. Classical Operator Splitting for Cahn–Hilliard

A paradigmatic instance is in the study of the Cahn–Hilliard equation, both with constant and variable mobility. For the scalar field $\phi$, the dynamics read: $$\phi_t = \nabla \cdot ( m(\phi) \nabla \mu ), \quad \mu = -\varepsilon^2 \Delta \phi + F'(\phi), \quad F(\phi) = \tfrac{1}{4} (\phi^2 - 1)^2,$$ with mobility $m(\phi)$. The Lie–Trotter operator splitting takes a time step $\tau$ and divides it into:

1. Nonlinear "reaction" step:

$$w \mapsto \widetilde{w} = w + \tau\,m(w)\Delta(f(w)), \quad f(w) = w^3 - w,$$

2. Linear "diffusion" step:

$$\widetilde{w} \mapsto w^{+} = e^{-\tau\,m(w)\Delta^2}\widetilde{w}.$$

The full time step is $\phi^{n+1} = S_L(\tau) S_N(\tau) \phi^n$, with explicit update formulas in both substeps. This scheme achieves unconditional energy stability, as quantified by the discrete $H^{-1}$-type energy

$$E_h(w) = \frac12 \|(-\Delta)^{-1/2}(e^{-\tau m \Delta^2} - I)w\|_{L^2}^2 + \int_{\Omega} F(w)\,dx,$$

satisfying $E_h(\phi^{n+1}) \leq E_h(\phi^n)$ for all $\tau > 0$ and $n \geq 0$ (Li et al., 2021). This property is distinct among splitting schemes for nonlinear, higher-order dissipative systems.

The procedure generalizes to variable mobility $m(\phi)$, higher-order PDEs, and nonlocal phase-field models by appropriate selection of linear and nonlinear substeps.

3. Mobility-Driven Variational Splitting for Cross-Diffusion

Recent advances introduce primal-dual forward-backward (PDFB) splitting in the minimization of variational formulations for cross-diffusion gradient flows with matrix mobility: $$\partial_t u = \nabla \cdot \left( M(u)\nabla \frac{\delta \mathcal{F}}{\delta u}(u) \right),$$ where $u\in \mathbb{R}^n$ and $M(u)\in \mathbb{R}^{n\times n}_{\mathrm{sym},+}$. The time-discretized minimizing-movement (JKO) step reads

$$u^{k+1} = \arg\min_u \mathcal{F}(u) + \frac{1}{2\tau} W_M^2(u, u^k),$$

with $W_M$ a distance induced by the mobility.

The PDFB framework (Deng et al., 31 Oct 2025) circumvents the full Benamou–Brenier time-continuous optimization by introducing dual fields and splitting the objective into prox-friendly and smooth components: $$\min_{x=(u, m)}\,\max_{y=(Q, q, \nu)}\,\Phi(x, y) - F(y) + G(x).$$ Here, the action of the mobility is (i) explicit in the coupling to dual variables $Q$ in $\Phi$, (ii) enforced in constraint sets (i.e., maintaining positive definiteness), and (iii) handled in the primal projection steps for $(u, m)$.

This approach provides practical and theoretically justified algorithms for systems with non-diagonal, even fully matrix-valued mobilities (e.g., Shigesada–Kawasaki–Teramoto cross-diffusion), preserving mass, positivity, and discrete energy dissipation at each iteration.

4. Mobility Splitting in Conservative-Dissipative and Measure-Valued Systems

Broader classes of PDEs, such as nonlocal, degenerate conservative-dissipative systems, also leverage operator splitting with mobility-structured steps (Adams et al., 2021). The generic form

$$\partial_t \rho + \nabla\cdot(\rho b[\rho]) = \nabla\cdot\big(A(\nabla\rho + \rho\nabla f)\big),$$

separates the conservative (transport) evolution from the dissipative (mobility-encoded Fokker–Planck) phase. The dissipative substep implements a variational move (often JKO/Wasserstein) with respect to a transport cost weighted by the mobility matrix.

Entropic regularization techniques can further accelerate this step by replacing the optimal transport with a Sinkhorn-regularized analogue.

A similar structure is present in measure-valued structured population models on metric spaces (Lindow et al., 11 Feb 2025). Here, the transport map (mobility) and the local/nonlocal reaction dynamics are handled sequentially. Implementation leverages explicit particle methods, where transport updates move Dirac masses according to the respective flow map, and reaction steps update masses via explicit Euler formulas, in both cases with careful control on the Lipschitz properties of model components.

5. Implementation Methodologies

The practical realization of mobility operator splitting is dictated by the structure of the equation and the computational domain. For Cahn–Hilliard-type equations, spectral methods in periodic domains allow diagonalization of the diffusion operator, significantly accelerating the linear substep. Nonlinear steps involve computation of nonlinearities (e.g., $f(w) = w^3-w$) followed by Poisson or Laplacian solves.

In variational (PDFB) settings, the splitting is tightly coupled to convex optimization primitives:

• Proximal updates on indicator or support functions of semidefinite cones (to maintain mobility positivity) are handled via projection or Newton methods.
• KKT systems in the primal projection are solved by active-set or saddle-point linear solvers.

For measure-valued settings on metric spaces, mobility splitting enables modular explicit schemes with clear error control. Spatial discretization involves finite-range approximation, and time discretization benefits from the theoretical splitting accuracy established for the semigroup structure of the dynamics (Lindow et al., 11 Feb 2025).

A summary table is provided to organize the main algorithmic steps for representative models:

Model Type	Splitting Sequence	Key Numerical Step
Cahn–Hilliard (constant mobility)	Nonlinear (reaction) → Linear (diffusion)	FFT-based Laplacian solve
Cross-diffusion (matrix mobility)	Variational minimization with PDFB splitting	Primal-dual convex–concave optimization
Conservative-dissipative (degenerate)	Transport (characteristics) → Dissipative (Fokker–Planck, mobility A)	Optimal transport / JKO or Sinkhorn minimization
Structured populations (measures)	Pushforward (mobility/transport) → Reaction/growth	Particle update, explicit Euler for masses

6. Convergence, Stability, and Structural Properties

Mobility-based operator splitting schemes, when properly constructed, preserve crucial structural properties of the underlying continuous dynamics:

• Unconditional energy stability: For Cahn–Hilliard, discrete energy $E_h$ is non-increasing for all time steps $\tau>0$.
• Uniform Sobolev bounds: For solutions with initial data in $H^k$, uniform-in-time bounds are available, dictated by mobility lower/upper bounds (Li et al., 2021).
• First-order time accuracy: Global $\mathcal{O}(\tau)$ accuracy is observed, with higher-order terms controlled by commutator estimates or smoothness of time-dependent coefficients (Lindow et al., 11 Feb 2025).
• Positivity and mass conservation: For gradient flows with primal–dual splitting, box constraints or indicator function projections preserve non-negativity and norm.
• Convergence for entropic-regularized and degenerate mobilities: Entropic regularization converges to the original PDE solution under mesh and regularization refinement (Adams et al., 2021).

A plausible implication is that the flexibility of mobility operator splitting makes it central for scalable and robust simulation of physically relevant nonlinear PDEs, especially when the mobility is degenerate, state-dependent, or fully coupled in systems.

7. Applications and Extensions

Mobility operator splitting finds application in:

• Phase separation and coarsening: Cahn–Hilliard and phase-field models with variable mobility;
• Complex multi-species cross-diffusion: Ecological, chemical, and crowded particle flows, as in the Shigesada–Kawasaki–Teramoto system;
• Kinetic equations: Fokker–Planck models, degenerate or nonlocal in nature;
• Structured population dynamics: Evolution of measures on abstract metric spaces, including Bayesian inverse problems where transport splitting is exploited to separate model parameter sensitivity;
• Nonlocal and higher-order flows: Extensions to thin-film epitaxy (6th order), phase-field-crystal models, and nonlocal interaction kernels.

In all instances, the operator splitting driven by mobility allows for modular, robust, and structure-preserving discretizations whose convergence and stability properties can often be proven rigorously, as seen in (Li et al., 2021, Deng et al., 31 Oct 2025, Adams et al., 2021), and (Lindow et al., 11 Feb 2025).