Maximum-Principle Schemes

• Maximum-principle-satisfying schemes are numerical methods that enforce discrete bounds in PDE solutions, preventing nonphysical oscillations.
• They employ techniques such as nonlinear rescaling limiters, flux corrections, and nodal projections to preserve high-order accuracy while maintaining physical constraints.
• These methods are pivotal in simulating scalar conservation laws, reaction-diffusion systems, and nonlocal or stochastic PDEs, ensuring both stability and robustness.

A maximum-principle-satisfying scheme (alternatively, maximum principle preserving or bound-preserving scheme) is a class of numerical methods specifically constructed to enforce discrete analogues of the maximum principle (MP), i.e., to ensure that numerical solutions never exceed physically or mathematically prescribed bounds, even at the fully discrete algebraic level. These schemes are foundational for robust, non-oscillatory, and physically credible simulation of a broad range of partial differential equations (PDEs) arising in scalar conservation laws, reaction-diffusion systems, phase field models, convection-diffusion equations, and coupled multi-physics phenomena, including those with strong nonlinearities, nonlocal terms, degeneracy, or under uncertainty.

1. Mathematical Framework and the Discrete Maximum Principle

At the continuous level, the maximum principle for a solution $u(x,t)$ to a prototypical parabolic, elliptic, or conservation law asserts that

$m \leq u(x,t) \leq M \quad \forall x, t,$

where $(m, M)$ are bounds often determined by initial/boundary data or physical constraints. Violations of this property at the numerical level can yield spurious oscillations, nonphysical solutions, or numerical instabilities.

A maximum-principle-satisfying numerical scheme constructs the discrete update so that, for all mesh cells (or nodes) and all time steps,

$u^n_i \in [m, M] \implies u^{n+1}_i \in [m, M],$

with $u^n_i$ representing the solution on the $i$-th cell at $n$-th time step. Preservation may be guaranteed under specific conditions (CFL-type or otherwise) or, for certain schemes, unconditionally.

2. Bound-Preserving Mechanisms and Limiter Designs

A central design challenge for high-order schemes lies in enforcing the maximum principle without degrading accuracy. Typical ingredients include:

• Nonlinear rescaling limiters: Apply post-processing to reconstructed polynomials (finite volume, finite element, or DG) to contract their excursions into $[m, M]$ while preserving cell averages and high-order accuracy, usually by a linear scaling with cell-specific amplification factors:

$$\tilde{u}_K(x) = \theta_K(u_K(x) - \bar{u}_K) + \bar{u}_K,$$

with $$\theta_K = \min\{1, \frac{M-\bar{u}_K}{M_K-\bar{u}_K}, \frac{\bar{u}_K-m}{\bar{u}_K-m_K}\}$$, $M_K$ and $m_K$ extremal values in cell $K$ (Guo et al., 2014, Yu et al., 2019, Friedrich et al., 2018).

• Cut-off nodewise projections: For finite element methods, a nodal post-processing step sets

$u_i^n := \max( \min( \hat{u}_i^n, M ), m )$

at each degree of freedom (Yang et al., 2021). This approach is trivial to implement and can be proven to preserve convergence rates for many time and space discretizations.

• Flux- or slope-limiting: For DG or FV methods, flux-correction transport (FCT)-type limiters or vertex-based slope limiters are used to enforce bounds for cell-averages and local extrema (Joshaghani et al., 2021, Luna et al., 2021). The approach identifies locally permissible increments via admissible fluxes and contracts any violating corrections.
• Design of monotonicity-preserving updates: Some schemes directly construct updates as convex combinations of neighboring admissible states, with all coefficients nonnegative and summing to unity, ensuring propagation of the bound (Chen et al., 2015, Yu et al., 2019, Chen et al., 2023).

3. Maximum-Principle-Satisfying Schemes Across Numerical Frameworks

Contemporary research demonstrates such schemes in diverse methodologies:

• Finite Volume WENO/Compact WENO with limiter: Compact high-order finite volume reconstructions coupled with stage-wise Zhang–Shu rescaling recover strict maximum-principle satisfaction for scalar conservation laws with arbitrary order in space (Guo et al., 2014).
• Finite Element and Lumped-Mass FEM: High-order lumped mass methods, possibly with arbitrarily high-order collocation-type time integrators, enforce bounds through nodal cut-off, often with optimal error estimates and optional energy-dissipation via scalar auxiliary variables (SAV) (Yang et al., 2021).
• Discontinuous Galerkin (DG): Both explicit and fully implicit high-order DG schemes on unstructured (including obtuse) meshes achieve cell-wise and pointwise bound preservation through a combination of carefully constructed numerical fluxes, scaling/vertex limiters, and monotonic time stepping (Chen et al., 2015, Yu et al., 2019, Joshaghani et al., 2021).
• Nonlocal and Fractional PDEs: Maximum-principle-satisfying finite-difference and exponential time-differencing (ETD) schemes are constructed for nonlocal, time-fractional, and space-fractional Allen–Cahn-type equations, using monotone discretizations and explicit step-size restrictions, or unconditional stability via stabilized splitting and matrix exponentials (Du et al., 2019, Zhang et al., 2023, Liao et al., 2019, He et al., 2018).
• Convection-Diffusion and Anisotropic Problems: For problems with variable-coefficient or highly anisotropic diffusion tensors—such as in plasma simulations—a hyperbolic system reformulation is exploited to enforce the DMP via spectral or algebraic constraints on algorithmic parameters, with no need for nonlinear limiting (Eto et al., 13 Aug 2025).
• Polynomial Chaos and Intrusive Polynomial Moment (IPM) Schemes: For conservation laws with stochastic input, strictly convex entropy-based closures allow rigorous definition and enforcement of maximum principles via bounded-barrier entropy and careful duality formulations, even in high-dimensional random spaces (Kusch et al., 2017).

4. Rigorous Analysis, Stability, and Accuracy

Analytical frameworks developed across these works are characterized by:

• Convex combination and monotonicity arguments: Discrete updates are shown to be monotone with respect to all input values under explicit, computable conditions, most notably via M-matrix (matrix with nonpositive off-diagonal entries and positive diagonal dominance) properties (Gute et al., 2021, Joshaghani et al., 2021).
• Proof of no order degradation: Limiting or cut-off operations are constructed to act only when needed (i.e., near sharp discontinuities or under resolved gradients), so high-order accuracy is retained in smooth regions—formalized via consistency, stability, and error-propagation theorems (Guo et al., 2014, Chen et al., 2015, Yang et al., 2021, Yu et al., 2019).
• No CFL restriction or unconditional stability: Many schemes are proven to satisfy the discrete maximum principle unconditionally (no time-step restriction), particularly for implicit time integrators (Luna et al., 2021, Du et al., 2019, Joshaghani et al., 2021). Where step sizes must be restricted, constraints are rigorous and computable.
• Energy dissipation laws at the discrete level: In phase field and Allen–Cahn models, schemes not only preserve the MP but also ensure the monotonic decay of discrete free energy under step-size constraints, mimicking the physical law (Chen et al., 2023, Zhang et al., 2023, Xu et al., 2020).

5. Extensions: Coupled, Nonlocal, Stochastic, and High-Dimensional Problems

Maximum-principle-satisfying methodology is documented across a broad spectrum:

• Coupled hydrodynamics and scalar transport: The preservation of steady states, positivity, and the discrete maximum-minimum principle for solute concentrations is crucial for accurate prediction over complex topography and moving-fluid domains (Karjoun et al., 2020).
• Long-range interactions and nonlocal gradient flows: The preservation of the maximum principle extends to nonlocal Cahn–Hilliard, Allen–Cahn–Ohta–Kawasaki, and fractional Allen–Cahn equations with consistent handling of nonlocal terms (Du et al., 2019, Xu et al., 2020).
• Uncertainty quantification via IPM: Schemes blending nonlinear entropy-principle-constrained closures and high-order spatial/temporal discretizations yield maximum-principle-satisfying solvers for conservation laws with random input, robust under both optimization inexactness and random-space coupling (Kusch et al., 2017).
• Nonuniform and adaptive time stepping: Advanced constructions handle memory effects and initial singularities in fractional PDEs via graded/aligned meshes and error-based adaptivity, all while rigorously keeping the discrete maximum (Liao et al., 2019).

6. Comparative Performance and Implementation Considerations

Numerical experiments reveal:

• Accuracy and robustness: Maximum-principle-satisfying schemes systematically eliminate spurious oscillations and nonphysical values at discontinuities, sharp interfaces, contact waves, and stochastic discontinuities (Guo et al., 2014, Yang et al., 2021, Kusch et al., 2017, Joshaghani et al., 2021).
• No accuracy sacrifice: Measured $L^1$, $L^2$, and $L^\infty$ errors confirm that high-order convergence rates are retained, even with aggressive limiting and on complex geometries or under mesh refinement (Guo et al., 2014, Yang et al., 2021, Chen et al., 2015, Friedrich et al., 2018).
• Versatility: The theory and algorithms encompass linear/nonlinear, local/nonlocal, isotropic/anisotropic, deterministic/stochastic, and time-fractional/standard problems, on arbitrary grids and with implicit/explicit stepping (Yu et al., 2019, Eto et al., 13 Aug 2025, Chen et al., 2023).
• Implementation: Many schemes can be realized with minor modifications to standard high-order solvers—by adding stage-wise post-processing, flux limiting, or local parameter tuning—without global solves or loss of parallel efficiency (Guo et al., 2014, Joshaghani et al., 2021, Luna et al., 2021).

7. Significance and Future Directions

Maximum-principle-satisfying schemes are now considered essential in the numerical analysis community for the robust simulation of strongly nonlinear, convective, and degenerate-diffusion dominated PDE systems. Advances in limiter design, mesh flexibility (e.g., with obtuse elements), adaptivity, and the handling of nonlocality and stochasticity have dramatically broadened the scope and applicability. Ongoing challenges include extending these strategies to fully coupled multi-physics, multiphase flows, adaptive and moving mesh algorithms, and high-dimensional stochastic systems, as well as deriving sharp conditions for unconditional preservation and optimizing computational performance in large-scale simulations (Guo et al., 2014, Yang et al., 2021, Liao et al., 2019, Yu et al., 2019, Joshaghani et al., 2021, Chen et al., 2023, Eto et al., 13 Aug 2025).