Compact 9-Point Finite Difference Method

• Compact 9-Point FDM is a 3×3 discretization framework using center, axial, and diagonal nodes to approximate 2D PDEs with high-order accuracy.
• It derives stencil coefficients systematically via Taylor expansions, recursive PDE reductions, and variable coefficient transformations.
• The method adapts to diverse settings—including variable-coefficient Poisson, convection-diffusion, and interface problems—ensuring stability through M-matrix properties.

A compact 9-point finite difference method is a two-dimensional $3\times 3$ finite difference discretization on a uniform Cartesian grid that uses the center, four axial neighbors, and four diagonal neighbors. In the compact, symmetric framework for the variable Poisson equation on a $d$-dimensional hypercube, compact means the stencil support is $S=\{-1,0,1\}^d$; in $d=2$, this becomes $S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$, which is exactly the Cartesian 9-point geometry (Feng et al., 4 Oct 2025). Across the recent literature, this geometry is used for variable-coefficient Poisson problems, convection-diffusion-reaction equations, nonlinear steady and unsteady problems, elliptic interface problems, and curved-domain discretizations, with different results for attainable order, symmetry, monotonicity, and convergence.

1. Definition, stencil geometry, and terminology

In its standard two-dimensional form, the compact 9-point method approximates a PDE at a grid point $(x_i,y_j)$ by coupling only the values at

$(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$

A generic operator is written as

$$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$

or, in index form,

$$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$

depending on the PDE normalization (Feng et al., 4 Oct 2025, Feng et al., 24 Jul 2025). The defining structural feature is that higher-order accuracy is sought without enlarging the local support beyond the $3\times 3$ neighborhood.

For symmetric Poisson-type constructions, the center coefficient is often constrained by

$d$0

so that constants are annihilated. For regular sixth-order elliptic constructions, the leading 9-point weight block is

$d$1

which is the standard compact 9-point Laplacian-like pattern corrected by higher-order $d$2-dependent terms involving derivatives of the coefficient field (Feng et al., 2023).

The term is not completely uniform across the literature. In optimized one-dimensional compact-scheme theory, a “9-point” central scheme corresponds to $d$3 and $d$4 nodes in a one-dimensional implicit stencil; in that setting, the phrase does not denote a $d$5 two-dimensional geometry (Deshpande et al., 2019). Other compact high-order papers are explicitly not classical compact 9-point FDMs: the jump-diffusion option-pricing method is a one-dimensional compact 3-point scheme (Patel et al., 2018), the third-derivative central compact scheme couples node and cell-center values on two interlaced grids rather than a single 9-node stencil (Salian et al., 2024), and the hybrid PINN paper uses the standard 5-point stencil for $d$6 in 2D while using compact finite differences only in one-dimensional subproblems (Xiang et al., 2022).

2. Construction principles, PDE reduction, and order barriers

The derivation of compact 9-point schemes in the current literature is systematic rather than ad hoc. For the variable Poisson equation

$d$7

one paper rewrites the PDE using

$d$8

so that

$d$9

This transformed form is then combined with Taylor expansion and consistency constraints to determine the stencil coefficients (Feng et al., 4 Oct 2025). For linearized convection-diffusion equations,

$S=\{-1,0,1\}^d$0

the derivation similarly uses recursive elimination of higher $S=\{-1,0,1\}^d$1-derivatives by the PDE, followed by local Taylor expansion and symbolic solution of the resulting algebraic conditions (Feng et al., 24 Jul 2025).

A representative elimination identity is

$S=\{-1,0,1\}^d$2

which is repeatedly applied until all higher derivatives are expressed in terms of a reduced derivative set and derivatives of the source term (Feng et al., 24 Jul 2025). For elliptic interface problems with variable coefficients, analogous recursive PDE reductions are expressed through $S=\{-1,0,1\}^d$3- and $S=\{-1,0,1\}^d$4-type basis polynomials and transmission formulas across the interface (Feng et al., 2023, Feng et al., 2022).

The attainable order depends sharply on the PDE class and on whether symmetry is imposed. For the variable Poisson equation on a uniform grid, a compact, symmetric 1D FDM can achieve arbitrary consistency order, but in dimensions $S=\{-1,0,1\}^d$5 the maximum consistency order that a compact, symmetric FDM on a uniform grid can achieve is $S=\{-1,0,1\}^d$6; if $S=\{-1,0,1\}^d$7 and the diffusion coefficient satisfies

$S=\{-1,0,1\}^d$8

then a compact symmetric sixth-order scheme exists (Feng et al., 4 Oct 2025). For the linearized convection-diffusion equation, the largest attainable order with a nontrivial compact 9-point stencil is $S=\{-1,0,1\}^d$9 (Feng et al., 24 Jul 2025). By contrast, in hybrid interface and curved-domain elliptic constructions, sixth-order compact 9-point stencils are obtained at regular interior points, with wider or modified stencils used only where the interface or boundary cuts the compact patch (Feng et al., 2023, Han et al., 17 Jan 2025).

3. Elliptic Poisson-type compact 9-point schemes

For the two-dimensional variable-coefficient Poisson problem,

$d=2$0

an explicit symmetric fourth-order compact 9-point scheme is given in midpoint form. With

$d=2$1

the nonzero off-center coefficients are

$d=2$2

$d=2$3

$d=2$4

$d=2$5

with

$d=2$6

and right-hand side

$d=2$7

This is the paper’s clearest explicit compact symmetric 9-point fourth-order formula in 2D (Feng et al., 4 Oct 2025).

If $d=2$8, then $d=2$9 is constant and the coefficients reduce to

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$0

The discrete equation becomes

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$1

which is the classical fourth-order 9-point Poisson formula (Feng et al., 4 Oct 2025).

Higher-order elliptic compact 9-point schemes remain active in variable-coefficient settings. For elliptic interface problems with mixed boundary conditions, the regular-point component of a hybrid method is a sixth-order compact 9-point stencil for

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$2

at interior regular points, and its leading coefficient block is again

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$3

The theorem states that all nontrivial solutions satisfy the sum condition for any $S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$4, and that there exists a nontrivial solution satisfying the sign condition for any $S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$5 (Feng et al., 2023).

4. Convection-diffusion-reaction, nonlinear equations, and time dependence

Compact 9-point schemes are also used for steady and unsteady convection-diffusion-reaction equations after rewriting the operator in the normalized form

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$6

A fourth-order explicit compact 9-point formula for this operator is

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$7

with short closed-form coefficients such as

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$8

$S=\{(0,0),\pm(1,0),\pm(0,1),\pm(1,1),\pm(1,-1)\}$9

The paper emphasizes that these simple explicit stencils satisfy the discrete maximum principle and form an M-matrix for the sufficiently small $(x_i,y_j)$0, if the function $(x_i,y_j)$1 is nonnegative (Feng, 17 Mar 2026).

For steady and time-dependent nonlinear convection-diffusion equations, the nonlinear PDE is first rewritten as a sequence of linear problems by fixed-point iteration. For the steady case,

$(x_i,y_j)$2

with

$(x_i,y_j)$3

and the paper derives a fourth-order compact 9-point finite difference method on a uniform Cartesian grid. It also gives a reduced-pollution-effect version for which

$(x_i,y_j)$4

For the time-dependent nonlinear equation, the temporal domain is discretized by Crank-Nicolson, BDF3, or BDF4; after linearization, the spatial problem again has the form

$(x_i,y_j)$5

and the overall fully discrete methods are reported as

$(x_i,y_j)$6

for CN, BDF3, and BDF4, respectively (Feng et al., 24 Jul 2025).

The same compact philosophy has been extended from rectangles to arbitrary curved domains. On smooth curved domains with Dirichlet data, a fourth-order compact FDM is used at every grid point on a uniform Cartesian mesh; at a regular stencil center the method uses the fourth-order compact 9-point FDM, while at an irregular stencil center it keeps only grid points inside $(x_i,y_j)$7 from the standard compact 9-point stencil and derives one of 10 case-dependent closures. For the left-hand side of the irregular stencil, one only needs to solve an at most $(x_i,y_j)$8 linear system, and the right-hand side is constructed in explicit expression (Feng et al., 20 May 2026).

5. Interfaces, irregular points, and curved-domain hybridization

The compact 9-point method is most effective when the full $(x_i,y_j)$9 patch lies in a single smooth region. Once an interface or boundary cuts the stencil, the literature generally changes the local formula rather than forcing a single uniform 9-point rule. For elliptic interface problems with discontinuous and high-contrast coefficients, the hybrid strategy is explicit: use a sixth-order 9-point scheme at interior regular points, a fifth-order 13-point scheme at irregular interior points, and compact boundary closures of smaller size (Feng et al., 2022, Feng et al., 2023).

Setting	Local stencil	Stated order or property
Interior regular point away from interface	9-point compact	Sixth-order (Feng et al., 2023, Feng et al., 2022)
Interior irregular point near interface	13-point	Fifth-order (Feng et al., 2023, Feng et al., 2022)
Boundary edge or corner on rectangle	6-point or 4-point	Sixth-order (Feng et al., 2023, Feng et al., 2022)
Smooth curved domain without ghost points	9-point interior; at most 8 inside-domain points near boundary	Sixth-order in $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$0 (Han et al., 17 Jan 2025)
Arbitrary curved domains with Cartesian truncation	9-point regular; truncated $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$1 near boundary	Fourth-order, 10 irregular cases (Feng et al., 20 May 2026)
Elliptic cross-interface problem	9-point fourth or sixth depending alignment	M-matrix in special aligned case (Feng et al., 2022)

For elliptic interface problems with mixed boundary conditions, the regular-point component of the hybrid method is a sixth-order compact 9-point stencil, while irregular points near the interface use a 13-point stencil and boundary points use sixth-order compact 6-point and 4-point schemes. The abstract states that, for the elliptic problem without interface, the compact FDM has the M-matrix property for any mesh size $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$2; near interfaces, the 13-point stencil is fifth-order consistent and the full hybrid method is reported to achieve global sixth-order convergence in the $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$3 norm (Feng et al., 2023).

For discontinuous and high-contrast coefficients, the same hybrid pattern reappears, but with a sharper negative statement: the paper states that the maximum accuracy order for compact 9-point finite difference scheme in irregular points is three. Its proposed remedy is an efficient hybrid implementation with sixth-order 9-point regular stencils, fifth-order 13-point irregular stencils, and sixth-order 6-point and 4-point boundary stencils (Feng et al., 2022).

Curved-domain work has pursued compactness without ghost points. One sixth-order paper for

$(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$4

in a two-dimensional curved domain proposes a sixth-order 9-point compact FDM that only utilizes the grid points in $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$5 for any mesh size $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$6, without relying on ghost points or information outside $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$7. All boundary stencils near $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$8 have at most 6 different configurations and use at most 8 grid points inside $(i+k,j+\ell),\qquad k,\ell\in\{-1,0,1\}.$9, and the paper proves

$$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$0

It also derives a gradient approximation directly from $$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$1 with order

$$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$2

in the $$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$3-norm for all $$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$4, with a logarithmic factor $$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$5 for $$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$6 (Han et al., 17 Jan 2025).

Cross-interface problems provide another specialized compact 9-point branch. For an elliptic problem with coefficient jumps across two intersecting straight internal interfaces, the paper proposes a fourth-order 9-point scheme in the general case, and, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, a compact 9-point scheme that can even reach sixth order of accuracy. In that special case, the resulting linear system has an M-matrix, and the theoretical sixth-order convergence rate is proved using the discrete maximum principle; in the general case, the paper also derives a compact third-order finite difference scheme yielding a linear system with an M-matrix (Feng et al., 2022).

6. Symmetry, M-matrices, fast solvers, and convergence theory

For Poisson-type compact schemes, symmetry is a major algebraic theme. In the midpoint formulation,

$$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$7

so the assembled matrix is symmetric by construction. For

$$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$8

with $$\mathcal L_h u_h(c)=\sum_{p\in S} C_p(c)\,u_h(c+ph),$$9, the continuous operator is coercive, and the paper states that the resulting linear system is in fact symmetric positive definite. This makes many fast solvers applicable and, because the stencil has minimum support, keeps the storage requirement minimal (Feng et al., 4 Oct 2025).

A second major theme is monotonicity. Many compact 9-point papers formulate sign and sum conditions of the form

$$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$0

or equivalently after sign reversal in the operator matrix. In the nonlinear convection-diffusion paper, the compact 9-point method generates an M-matrix, provided the mesh size $$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$1 is sufficiently small, and all FDMs satisfy the discrete maximum principle for sufficiently small $$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$2 (Feng et al., 24 Jul 2025). In the variable-coefficient convection-diffusion-reaction paper, each proposed 2D and 3D compact scheme satisfies the discrete maximum principle and forms an M-matrix for the sufficiently small $$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$3, if the function $$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$4 is nonnegative (Feng, 17 Mar 2026).

Some compact 9-point results are stronger. For sixth-order hybrid finite difference methods for elliptic interface problems with mixed boundary conditions, the regular-point compact 9-point stencil satisfies sign and sum conditions ensuring the M-matrix property for any mesh size $$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$5 in the non-interface problem (Feng et al., 2023). For the constant-coefficient transport problem on the unit square, a sixth-order compact 9-point FDM forms an M-matrix for any mesh size $$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$6, and the paper proves

$$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$7

by a discrete maximum principle using the explicit comparison function

$$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$8

for any $$h^{-2}\mathcal L_h u_h=h^{-2}\sum_{k=-1}^1\sum_{\ell=-1}^1 C_{k,\ell}(u_h)_{i+k,j+\ell}=F_{i,j},$$9 (Feng, 14 Jun 2025).

Convergence proofs for compact 9-point methods therefore split into two main regimes. In symmetric coercive Poisson problems, SPD structure supports Krylov and fast elliptic solvers (Feng et al., 4 Oct 2025). In monotone interface or transport problems, the decisive tool is the discrete maximum principle. This is how sixth-order $3\times 3$0-norm convergence is proved for the aligned cross-interface compact 9-point method (Feng et al., 2022), for the constant-coefficient sixth-order transport stencil (Feng, 14 Jun 2025), and for the curved-domain sixth-order compact method without ghost points (Han et al., 17 Jan 2025). A practical implication is that the compact 9-point method is not a single scheme class but a family whose core tradeoffs are local support versus attainable order, and symmetry/SPD versus sign-structured M-matrix behavior, depending on the PDE and the geometry.