Monotone Numerical Schemes

• Monotone numerical schemes are discretization methods for PDEs that enforce order-preservation using nonnegative stencil coefficients and M-matrix properties.
• They guarantee stability and convergence by upholding discrete maximum principles and comparison properties, even for degenerate or non-smooth problems.
• Widely used in conservation laws, control problems, and fractional PDEs, these schemes balance robustness with lower formal accuracy compared to non-monotone methods.

Monotone numerical schemes are a class of discretization methods for partial differential equations (PDEs), difference equations, and related operator problems that possess strong discrete analogues to maximum principles and order-preservation properties. The central principle is that the scheme’s update operator is non-decreasing with respect to its primary argument and non-increasing with respect to coupling terms, so that the discrete system preserves monotonicity, comparison principles, and often positivity. This property is pivotal for stability, convergence to viscosity solutions, and robustness on non-smooth, degenerate, or highly nonlinear problems. Monotone schemes arise throughout conservation laws, Hamilton-Jacobi-Bellman equations, degenerate elliptic and parabolic equations, nonlocal problems, and more.

1. Structural Definition and Core Principles

A monotone scheme is one for which, given an update formula

$u_i^{n+1} = F(\{u_j^n\}_{j\in S}),$

the map $F$ is non-decreasing in $u_i^n$ (the central node) and non-increasing in neighboring nodes, guaranteeing that the discrete operator is monotone in the sense of order-preserving mappings. In the context of finite-difference and finite-volume schemes, this corresponds to nonnegative stencil coefficients off the diagonal and a positive diagonal, such that the assembled matrix is an M-matrix—irreducible, diagonally-dominant, and with non-positive off-diagonals.

Monotonicity is crucial for:

• Discrete maximum principles (DMP) and local extrema diminishing (LED) properties.
• Stability under nonlinearities and degeneracies.
• Comparison principles, ensuring that sub-solutions stay beneath super-solutions at every node.
• Convergence to viscosity solutions in non-smooth, non-convex, or path-dependent regimes (Chen et al., 2016, Ren et al., 2015, Debrabant et al., 2014).

The Barles–Souganidis framework formalizes that monotonicity, consistency, and stability together ensure uniform convergence of the scheme to the viscosity solution of the underlying PDE, including fully nonlinear, nonlocal, and path-dependent equations.

2. Applications: Conservation Laws and Nonlinear Equations

Monotone schemes are foundational for scalar conservation laws with discontinuities:

$u_t + f(u)_x = 0,$

where monotone finite-volume schemes—including Lax–Friedrichs, Engquist–Osher, and Godunov—provide discrete solutions that are $W_1$-contractive. Remarkably, these first-order schemes can achieve second-order convergence rates in the Wasserstein distance $W_1$ for piecewise-constant, decreasing initial data, far outperforming their classical $L^1$ error rates (Fjordholm et al., 2016). The mechanism is that $W_1$ measures mass transport ("mass × distance"), so smeared shocks of width $O(\Delta x)$ contribute only $O(\Delta x^2)$ in $W_1$.

For strongly nonlinear degenerate equations—including the Monge–Ampère equation and fully nonlinear Bellman–Isaacs equations—monotonicity is enforced by wide stencil semi-Lagrangian schemes, nonnegative directional splittings, or reformulation as Hamilton–Jacobi–Bellman PDEs. Schemes switch adaptively between compact stencils (when monotonicity holds) and more diffuse, unconditional monotone stencils otherwise, maintaining convergence guarantees even in singular regimes (Debrabant et al., 2014, Chen et al., 2016, Ngo et al., 2015).

3. Monotonicity-Preserving Construction and Discretization Techniques

The design of monotone schemes typically involves:

• Careful selection of upwind or central differences, ensuring nonnegative weights and satisfaction of discrete maximum principles.
• Nonnegative directional splitting: decomposition of anisotropic operators into locally nonnegative 1-D diffusion terms, potentially requiring larger stencils for more anisotropic problems (Ngo et al., 2015).
• Adaptive choice between standard narrow stencils and semi-Lagrangian wide stencils, exploiting convexity or diagonal-dominance where available (Chen et al., 2016).
• Use of monotone interpolation (e.g., tent functions or positive linear basis) in schemes for Bellman–Isaacs and control problems, guaranteeing monotonicity for arbitrary meshes (Debrabant et al., 2014).
• Artificial diffusion and shock detectors (for dG methods), with graph-Laplacian contributions ensuring monotonicity on general and unstructured meshes (Badia et al., 2016).

In nonlocal and fractional equations, monotonicity is preserved by positive quadrature schemes and upwind-modified drifts, allowing for rigorous error rates that track the order of the nonlocal operator, the regularity, and the nature of the kernel dependence (Biswas et al., 2017).

4. Convergence, Stability, and Comparison Principles

Monotonicity is entwined with convergence and stability:

• Any monotone, consistent, stable scheme converges to the unique viscosity solution under the Barles–Souganidis theory, regardless of smoothness or degeneracy (Ren et al., 2015, Chen et al., 2016).
• Monotonicity ensures discrete comparison principles, critical for path-dependent PDEs and stochastic control (via monotone discretizations of nonlinear expectations), which hold even for non-Markovian and backward stochastic differential equations (Ren et al., 2015).
• In time-discrete schemes for gradient flows and phase-field models, monotonicity is more decisive than energy stability for guaranteeing correct asymptotic behavior; only implicit Euler admits a uniform step-size threshold ensuring monotonic convergence for arbitrary initial data (Li et al., 2024).

Additionally, monotone schemes exhibit robust stability properties: total-variation diminishing for conservation laws, discrete maximum principles for reaction-diffusion, and norm-preserving dynamics for fractional ODEs via $\mathcal{CM}$-preserving convolution kernels (Li et al., 2019).

5. Error Analysis and Convergence Rates

Error analysis for monotone schemes often reveals precise, algebraic rates linked to the underlying operator structure and regularity:

• For discontinuous solutions of scalar conservation laws, monotone schemes yield $W_1$ error $O(\Delta x^2)$, while $L^1$ error remains only $O(\Delta x)$ (Fjordholm et al., 2016).
• In problems with spatially discontinuous flux, monotone schemes are proven to converge at $O(\sqrt{\Delta x})$ in $L^1$ (Badwaik et al., 2019).
• Robust error bounds exist in nonlocal and fractional PDEs, including $O(\Delta x^{2-\sigma})$ rates for nonlocal equations of fractional order $\sigma$ (Biswas et al., 2017).
• For fully nonlinear path-dependent or high-dimensional Bellman–Isaacs equations, monotone schemes attain rates tied to grid resolution, moment matching, and regularity of the generator (Guo et al., 2012, Ren et al., 2015).
• In monotone anisotropic diffusion, required stencil sizes scale with anisotropy, but second-order convergence is attainable when monotone splitting applies (Ngo et al., 2015).

The table below summarizes representative rates from key application areas. | Scheme/Class | Problem Type | Proven Rate (norm/domain) | |----------------------|---------------------|---------------------------| | Lax–Friedrichs, Godunov | Scalar Cons. Law, convex flux | $O(\Delta x^2)$ in $W_1$, $O(\Delta x)$ in $L^1$ (Fjordholm et al., 2016) | | Monotone FV, upwind | Discontinuous flux | $O(\sqrt{\Delta x})$ in $L^1$ (Badwaik et al., 2019) | | Nonlocal finite-diff. | Fractional Isaacs | $O(\Delta x^{2-\sigma})$ (Biswas et al., 2017) | | Adaptive mixed-stencil| Monge–Ampère | $O(h^2)$ (narrow stencil), $O(h)$ (wide) (Chen et al., 2016) | | Nonneg. directional FD| Aniso. diffusion | $O(h^2)$ (Ngo et al., 2015) |

6. Extensions, Generalizations, and High-Dimensional Problems

Monotone schemes generalize across PDE types and application domains:

• Path-dependent PDEs and non-Markovian stochastic control admit unified monotone convergence theorems, covering finite-difference, semi-Lagrangian, probabilistic, and controlled Markov-chain discretizations (Ren et al., 2015).
• High-dimensional fully nonlinear parabolic PDEs utilize monotone trinomial kernels and positive interpolation, allowing practical schemes up to dimension 12 (Guo et al., 2012).
• Nonconvex regularization functionals (flexible sparsity) can be addressed with monotone majorization–minimization algorithms that provably decrease objective values and converge robustly under M-matrix structure (Ghilli et al., 2021).
• Block-monotone iteration methods (Jacobi/Gauss–Seidel) guarantee existence, uniqueness, and bracketing of solutions in large-scale nonlinear systems with quasi-monotone coupling (Al-Sultani, 2019).
• Two-scale monotone schemes in integrodifferential, nonlocal, and free-boundary problems combine coarse regularization and fine mesh monotonicity, enabling maximum principles and sharp boundary error estimates (Borthagaray et al., 2024).

7. Methodological Implications and Practical Guidelines

Monotone numerical schemes, through their order-preserving structure:

• Enable stable, convergent approximations in regimes of low regularity, strong degeneracy, or nonlinearity, where standard high-order methods may be unstable or non-convergent.
• Require careful discretization and, in many cases, larger stencils or adaptive switching to maintain monotonicity—especially in highly anisotropic or nonlocal problems.
• May have lower formal accuracy compared to non-monotone, high-order schemes, but produce solutions that rigorously approximate the correct viscosity solution and respect fundamental physical constraints (e.g., positivity, boundedness).
• Are essential for advancing numerical analysis in stochastic control, differential games, phase-field models, fractional PDEs, and singular nonlinear problems.

A plausible implication is that the expansion of monotonicity-preserving techniques to high-order, high-dimensional, or complexly coupled systems remains an active area of research. Methodological innovations continue to address practical computational challenges, including efficiency, adaptive grid control, and monotonicity enforcement on general meshes or arbitrary domains.