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Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis
So-Hsiang Chou, Patrick Nyadjo Fonga
TL;DR
Interface schemes can retain second-order accuracy even when interface-adjacent truncation errors are O(1), a behavior not explained by pointwise consistency alone. The paper analyzes conservative flux-form operators through explicit discrete Green’s functions, showing that flux balance cancels neighboring defects and yields second-order maximum-norm convergence for a scheme class including weighted harmonic discretization. The mechanism also persists along grid lines for flat interfaces on Cartesian grids.
Problem
Interface finite-difference schemes may have O(1) local errors near discontinuities while achieving second-order accuracy, and pointwise consistency alone does not explain this behavior.
Method
The paper derives an explicit discrete Green’s function for a conservative flux-form operator and uses its weighted-increment structure to analyze cancellation of interface defects.
Results
The admissible conservative interface-scheme family achieves second-order global convergence in the maximum norm, including the weighted harmonic discretization.
Takeaways & Limitations
Second-order accuracy can result from conservative cancellation and Green-function propagation of localized errors rather than second-order pointwise consistency everywhere.
Takeaways & Limitations
The paper does not pursue a complete maximum-norm convergence analysis for the two-dimensional case because the required Green-function step does not carry over directly.
Abstract
from arXiv · showhide
Finite difference methods for interface problems often exhibit large local truncation errors near the interface, particularly when discontinuities in coefficients or solution derivatives are present. Nevertheless, many such schemes achieve second-order accuracy, a phenomenon not fully explained by standard pointwise consistency arguments. In this work, we provide a precise explanation of this behavior through the structure of the discrete Green's function associated with the underlying conservative difference operator. An explicit representation of the Green's function reveals a two-plateau structure in its weighted increments. By expressing the numerical error in terms of this Green kernel, we show that the dominant interface truncation errors, although individually of order \(O(1)\), possess a cancellation structure that reduces their effective contribution to the global error. The exact Green's-function analysis leads to a class of conservative interface flux--balance schemes for which the cancellation mechanism yields second-order accuracy in the maximum norm. The weighted harmonic discretization is included as a particular member, and its cancellation property is established directly. For Cartesian grids with a flat interface parallel to a coordinate direction, the same conservative cancellation mechanism persists along grid lines crossing the interface. Numerical experiments in one and two dimensions directly illustrate the predicted cancellation behavior and the resulting second-order accuracy. These results demonstrate that second-order accuracy in interface problems can arise from the interaction between localized truncation errors and the global structure of the discrete operator, rather than from pointwise consistency alone.
1. Introduction
The paper explains second-order accuracy despite O(1) interface truncation errors by combining conservative flux balance with the discrete Green’s function structure. It develops a general conservative scheme class, includes the weighted harmonic discretization, and extends the cancellation mechanism to flat Cartesian interfaces.
- Motivation: Interface-adjacent truncation errors can remain O(1), so pointwise consistency alone does not explain schemes’ second-order global accuracy.The issue arises especially with discontinuous coefficients or solution derivatives.
- Discrete Green’s function: An explicit inverse formula for the conservative flux-form operator reveals two constant plateaus in each column’s weighted Green-function increments.The scaled operator also satisfies a mesh-independent maximum-norm stability estimate.
- Cancellation mechanism: Two individually O(1) interface defects can contribute only O(h2) to the global error through the Green-function representation.The Green kernel determines how the localized, cancellation-structured defects propagate through the discrete solution.
- Cancellation mechanism: Conservation makes the neighboring interface equations share a cut-face flux with opposite signs, so their leading interface defects cancel when the equations are combined.The combined consistency error is one order smaller than the individual defects.
- Scheme class: Every admissible conservative interface scheme in the proposed family achieves second-order convergence in the maximum norm.The family requires positive face coefficients, a common cut-face flux, O(1) individual defects, and a combined consistency condition.
- Extensions: The weighted harmonic discretization is a member of the family, and on Cartesian grids with a flat interface the mechanism extends row by row or column by column.A complete multidimensional Green-function analysis is identified as an additional issue.
2. Discrete Green’s Function and Error Representation
The conservative flux-form operator has an explicit discrete Green’s kernel whose weighted increments form two constant plateaus. This structure yields useful bounds for representing and estimating the numerical error.
- The one-dimensional operator is a scaled stiffness matrix for the positive flux-form operator −(βu_x)_x, with positive face coefficients.
- The inverse of the flux-form matrix has an explicit representation based on cumulative reciprocal conductivities.
- For each source index, the weighted Green’s increments take exactly two constant values, separated at the source node.
- This two-plateau property is independent of the interface location and averaging rule, following from the conservative tridiagonal structure.
- The scaled Green’s kernel satisfies G_ij = O(h), G_i+1,j − G_ij = O(h^2), and analogous bounds in the second index.
- The interface truncation errors satisfy τ_I = O(1), τ_I+1 = O(1), and τ_I + τ_I+1 = O(h).
- Green-kernel weighting converts these two large defects into an O(h^2) contribution to the global error.
3. Stability and Conditioning
The scaled operator has mesh-independent maximum-norm stability under a fixed lower ellipticity bound, while spectral conditioning worsens with mesh refinement and coefficient contrast.
- The maximum-norm stability estimate is independent of the mesh size.
- The mesh-independent inverse bound applies to the scaled operator L_h, not the unscaled stiffness matrix A_h.
- The stability bound depends only on the lower ellipticity bound, so increasing conductivities does not destroy uniform stability.
- The algebraic spectral condition number grows like h^-2 and may also deteriorate with coefficient contrast.
- Positive cut-face coefficients preserve the symmetric positive definite conservative flux form used in the analysis.
4. Large Local Errors and Their Cancellation
Second-order global accuracy can persist despite O(1) interface truncation errors when their leading terms cancel. The Green’s-function estimates convert this cancellation into an O(h^2) maximum-norm error.
- Second-order global accuracy requires cancellation of the two large interface defects, not second-order local consistency at both adjacent nodes.
- For the weighted harmonic scheme, the adjacent defects satisfy τ_I = O(1), τ_I+1 = O(1), and τ_I + τ_I+1 = O(h).
- The Green’s-function decomposition gives G_iI τ_I + G_i,I+1 τ_I+1 = O(h^2).
- Regular nodes contribute O(h^2), and the interface contribution is also O(h^2), yielding second-order maximum-norm convergence.
- The weighted harmonic interface scheme therefore satisfies the same second-order estimate.
- The mechanism combines conservative flux structure, two-plateau Green increments, and cancellation of the leading interface defects.
5. A Class of Conservative Interface Schemes
The paper defines admissible conservative interface schemes through shared cut-face fluxes, positive face coefficients, and local and combined consistency conditions. Every scheme in this class achieves second-order maximum-norm convergence.
- The two interface-adjacent equations share one numerical cut-face flux with opposite signs.
- Adding the adjacent equations cancels the common cut-face flux exactly, expressing the scheme’s conservation property.
- Admissible schemes require positive face coefficients and the conservative flux form of the underlying operator.
- Every admissible interface scheme satisfies second-order global convergence in the maximum norm.
- The weighted harmonic discretization is an admissible member, while the analysis also covers alternatives preserving the same structural conditions.
6. The Cancellation Mechanism in Two Dimensions
For a flat interface aligned with a Cartesian coordinate direction, the one-dimensional conservative cancellation mechanism extends along grid lines, while a complete multidimensional Green’s-function proof remains open.
- Setup: The analysis targets a flat vertical interface and applies the one-dimensional conservative interface discretization independently on each horizontal grid line.The interface location and cut geometry are identical across horizontal lines, with jump corrections entering only the normal-direction flux.
- Row-wise cancellation: Each pair of interface-adjacent equations shares one cut-edge flux, which cancels exactly when the equations are added.The same mechanism applies with x and y interchanged for a horizontal interface.
- Row-wise cancellation: Under uniformly bounded one-sided derivatives and admissible one-dimensional schemes on every row, the row-wise cancellation estimate holds uniformly along the interface.The proof decomposes truncation error by coordinate direction: the crossing direction uses one-dimensional cancellation, while the tangential direction remains a smooth-domain centered discretization.
- Scope: The two-dimensional extension identifies conservative flux balance and neighboring-defect cancellation as surviving ingredients, but not the complete Green’s-function step.A different two-dimensional discrete Green’s kernel would require new estimates for a full maximum-norm convergence analysis.
- Scope: For curved interfaces, varying intersections, non-coordinate normals, and geometrically consistent distribution of normal-flux jumps introduce additional unresolved geometric issues.A rigorous theory would additionally require geometric consistency estimates and multidimensional discrete Green’s-function bounds.
7. Numerical illustration of the cancellation mechanism
One- and two-dimensional experiments directly compare large interface-adjacent defects with their combined effect and observe second-order maximum-norm convergence in the tested settings.
- Experimental purpose: The experiments are designed to display the contrast between large individual interface truncation errors and their much smaller combined effect.They are illustrative rather than an extensive numerical comparison.
- 7.1. One-dimensional example.: In one dimension, the two interface defects remain O(1), while their normalized combined defect remains bounded under refinement.Their magnitudes alternate between two families because the fixed interface changes relative position within the cut interval on dyadic meshes.
- 7.1. One-dimensional example.: One-dimensional maximum-norm errors measured along fixed interface-position subsequences are consistent with ∥U − uh∥∞ = O(h2).For N = 64, 256, 1024, reducing h by a factor of four produces reductions consistent with second-order convergence.
- 7.2. Two-dimensional example.: In two dimensions, row-wise interface defects remain bounded despite marked alternation with the interface’s relative mesh position.The bounded combined defects agree with the row-wise cancellation estimate.
- 7.2. Two-dimensional example.: Two-dimensional maximum-norm errors along fixed interface-position subsequences show reductions consistent with second-order convergence.For N = 32, 128, 512, h is reduced by a factor of four along each subsequence.
- 7.2. Two-dimensional example.: The two-dimensional computation illustrates row-wise cancellation, but a complete two-dimensional Green’s-function proof of the global result is beyond the analysis.The test uses a vertical interface parallel to a coordinate direction and homogeneous boundary conditions.
Appendix A. Interface Truncation-Error Cancellation
The appendix proves that interface-adjacent truncation errors can each be O(1) while their sum is O(h), establishing the local cancellation relation used by the Green’s-function analysis.
- Interface geometry: The interface geometry is parameterized by δℓ = α − xI and δr = xI+1 − α, with δℓ + δr = h.The relative position can be written as δℓ = θh and δr = (1 − θ)h.
- Weighted harmonic discretization: The weighted harmonic coefficient uses the interface subinterval lengths and agrees with the coefficient defined in Lemma 4.2.The corrected cut-interval flux is rewritten in symmetric matrix form, transferring correction terms to the right-hand side.
- Individual defects: Taylor expansions at the left and right interface nodes show that potentially singular terms cancel exactly on each side’s local estimate.The resulting individual truncation errors satisfy τI = O(1) and τI+1 = O(1).
- Combined defect: Adding the two truncation errors eliminates the common cut-interval flux exactly, while the jump terms cancel in the combined relation.The combined estimate uses centered flux and source-term expansions on the smooth subintervals.
- Combined defect: The final estimates are τI = O(1), τI+1 = O(1), and τI + τI+1 = O(h).This is the cancellation relation required for the interface Green’s-function argument.