MAC Conservative Finite-Difference Method

• The MAC conservative finite-difference approach is a spatial discretization method that employs staggered grids to enforce local conservation in simulating fluid flow and elasticity.
• It utilizes specialized finite-difference operators and staggered unknown placements to achieve second-order accuracy, super-convergence, and robustness against locking in nearly incompressible regimes.
• Extensive numerical experiments demonstrate reliable convergence on both uniform and non-uniform grids, underpinning its effectiveness in high-fidelity simulations of complex PDEs.

A marker-and-cell (MAC) conservative finite-difference approach is a class of spatial discretization schemes for partial differential equations (PDEs) on structured grids, featuring staggered arrangements of primary variables to enforce local conservation laws and suppress spurious oscillations. These strategies are foundational in numerical fluid mechanics and elasticity, underpinning accurate and robust simulations of incompressible flow, Stokes problems, and linear elasticity on regular or non-uniform grids. The MAC approach centers on the placement of vector components at cell faces or edges and scalar quantities at cell centers, providing a geometric structure that ensures discrete analogues of fundamental conservation and constitutive relations.

1. Staggered Grid Layout and Unknown Placement

The marker-and-cell strategy employs staggered grids to allocate unknowns according to the underlying physical PDE. For incompressible flow and pressure–velocity systems (e.g., the steady Stokes equations), the classical layout positions velocity components on cell faces and pressures at cell centers:

• $u$ (horizontal velocity) at east–west faces $(x_{i+\frac{1}{2}}, y_j)$,
• $v$ (vertical velocity) at north–south faces $(x_i, y_{j+\frac{1}{2}})$,
• $p$ at cell centers $(x_{i+\frac{1}{2}}, y_{j+\frac{1}{2}})$.

In the treatment of linear elasticity in the displacement-stress formulation, the MAC-E scheme extends these principles:

• Displacement components $u^x$ at midpoints of vertical edges $(i, j+\frac{1}{2})$,
• Displacement components $u^y$ at midpoints of horizontal edges $(i+\frac{1}{2}, j)$,
• Normal stresses $\sigma^{11}, \sigma^{22}$ at cell centers $(i+\frac{1}{2}, j+\frac{1}{2})$,
• Shear stresses $\sigma^{12}$ at primary grid nodes $(i, j)$.

This placement preserves the geometric conservation structure of the original PDEs by matching discrete unknowns to the natural locations of continuity and equilibrium balances (Rui et al., 12 Jan 2026, Blinkov et al., 2018).

2. Conservative Finite-Difference Operators

Discretization proceeds via finite-difference approximations for derivatives, constructed to maintain local conservation. On uniform meshes, central differences provide second-order accuracy; on non-uniform meshes, operators are adapted:

• Forward difference for $M \rightarrow TM$: $$d_x\phi_{i+\frac{1}{2}, m} = \frac{\phi_{i+1, m} - \phi_{i, m}}{h_{i+\frac{1}{2}}}$$,
• Central/backward difference for $TM \rightarrow M$: $$D_x\phi_{i, m} = \frac{\phi_{i+\frac{1}{2}, m} - \phi_{i-\frac{1}{2}, m}}{h_i}$$,
• Analogous operators in $y$.

For the MAC scheme in incompressible flow, five-point stencils represent the Laplacians and divergences:

$$\Delta_1(\phi) = \frac{\phi_E + \phi_N - 4\phi_0 + \phi_S + \phi_W}{h^2},$$

where $\phi$ is evaluated at staggered positions. The pressure–Poisson equation on the coarser $2h$ grid employs a similar five-point stencil (Blinkov et al., 2018).

3. Discrete Equations: Constitutive, Momentum, and Conservation Laws

The core of the conservative MAC method is a discrete system that enforces constitutive laws, momentum balances, and divergence constraints at the grid scale. For linear elasticity in displacement–stress form (MAC-E), the discrete scheme includes:

• Constitutive equations for normal and shear stress, discretized at respective locations, e.g.:

$$\frac{1}{2\mu}\left(Z^{11}_{i+\frac{1}{2}, j+\frac{1}{2}} - \frac{\lambda}{2\lambda + 2\mu}(Z^{11} + Z^{22})_{i+\frac{1}{2}, j+\frac{1}{2}}\right) - d_x W^x_{i+\frac{1}{2}, j+\frac{1}{2}} = 0$$

• Shear stress-conjugate displacement equation:

$$Z^{12}_{i,j} - \mu (D_y W^x_{i,j} + D_x W^y_{i,j}) = 0$$

• Discrete equilibrium (divergence):

$$D_x Z^{11}_{i, j+\frac{1}{2}} + d_y Z^{12}_{i, j+\frac{1}{2}} = f^x_{i, j+\frac{1}{2}}$$

For incompressible flow, the discrete divergence and pressure–velocity coupling are imposed on the corresponding control volumes:

$$\frac{u_{i+\frac{1}{2},j} - u_{i-\frac{1}{2},j}}{h} + \frac{v_{i,j+\frac{1}{2}} - v_{i,j-\frac{1}{2}}}{h} = 0$$

These arrangements guarantee that conservation constraints are satisfied pointwise by the discrete solution (Rui et al., 12 Jan 2026, Blinkov et al., 2018).

4. Stability, Consistency, and Super-Convergence

Theoretical analysis of the MAC schemes applies discrete analogues of functional analysis to establish well-posedness, stability, and accuracy:

• Stability: The MAC-E scheme achieves stability via a discrete inf–sup condition (Ladyzhenskaya–Babuška–Brezzi) and coercivity on the kernel of the coupling form. Constants in the stability estimate depend only on the shear modulus $\mu$, not on the Lame parameter $\lambda$—hence, the approach is locking-free and robust as $\lambda \rightarrow \infty$ (nearly incompressible regime).
• Strong Consistency: For Stokes flow, the conservative scheme’s strong consistency is demonstrated by constructing the difference Janet/Gröbner basis of the scheme’s discrete ideal and matching every consequence to the continuous PDEs in the zero-mesh limit. There are no spurious (nonphysical) difference equations introduced at the discrete level, ensuring algebraic fidelity (Blinkov et al., 2018).
• Super-Convergence: The MAC-E scheme provides $O(h^2)$ super-convergence in $L^2$ for both displacement and stress by patch interpolation and Taylor-expansion analysis. The error remains second-order accurate for all unknowns, regardless of parameter degeneracy, with the error bound

$$\|W^x - u^x\|_{TM} + \|W^y - u^y\|_{MT} + \|Z^{ij} - \sigma^{ij}\| = O(h^2 + \ell^2)$$

for mesh sizes $h, \ell$ (Rui et al., 12 Jan 2026).

5. Numerical Results and Practical Performance

Extensive numerical experiments corroborate the theoretical properties:

• 2D Elasticity: Both uniform and non-uniform grids, and compressible and nearly incompressible regimes, achieve observed rates near $2.0$ in all discrete $L^2$-norms.
• 3D Elasticity: Uniform grid refinement yields second-order convergence, robust to the nearly incompressible limit.
• Steady Stokes Flow: The strongly consistent scheme allows larger grid spacing (about $1.7 \times$ coarser than classical MAC) before surpassing a fixed relative error in velocity, with potential $2-3\times$ speed-up at fixed accuracy (Blinkov et al., 2018).

Key performance characteristics are summarized below:

Property	MAC-E (Elasticity)	Strongly Consistent Stokes Scheme
Grid	Non-uniform, staggered	Uniform, staggered
Conservation	Local equilibrium enforced	Local via control volumes
Convergence	$O(h^2)$, super-convergent	$O(h^2)$, second-order
Locking	Locking-free	Not applicable
Consistency	Standard, via error analysis	Strong via Janet/Gröbner basis

6. Comparison with Classical MAC and Extensions

Both the classical MAC method and its conservative finite-difference extensions share key geometric and algebraic foundations:

• Staggered Arrangements: Both place vector and scalar unknowns to align with control volumes of the divergence and balance equations.
• Central Differencing: Both utilize second-order centered stencils for discreteness and accuracy.
• Conservative Structure: Each discrete equation represents an integral physical conservation statement over the associated control volume.
• Pressure–Velocity Coupling: Classical MAC often requires projection or fractional-step procedures; the strongly consistent Stokes scheme includes the pressure-Poisson coupling directly.
• Boundary Closure: Both approaches systematically enforce Dirichlet or Neumann boundary conditions consistent with the primary unknowns’ locations.

Distinctly, the strongly consistent schemes derived via symbolic difference elimination (Janet/Groebner) guarantee algebraic inheritance of all continuous constraints without hidden artifacts, a property not generally proved for classical ad hoc MAC discretizations (Blinkov et al., 2018).

7. Significance, Implementational Practices, and Scope

The MAC conservative finite-difference approach remains a central scheme in computational mechanics due to its discrete conservation, stability under inf–sup constraints, super-convergent error behavior, and robustness to locking phenomena. The symbolic formalism leveraging difference ideals and contoured volume integrals extends its reliability to more complex systems and ensures direct algebraic mapping between discrete and continuous problem statements.

A plausible implication is that these conservative, strongly consistent, and locking-free variants provide a preferred basis for high-fidelity numerical simulations in both fluid and solid mechanics, particularly for applications requiring strict conservation, uniform grid convergence, and strong algebraic guarantees (Rui et al., 12 Jan 2026, Blinkov et al., 2018). Many standard MAC-based simulation codes can be adapted with modest structural changes to implement the enhanced conservative and s-consistent stencils, offering improved fidelity and efficiency in demanding computational settings.