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High-order, Compact, and Symmetric Finite Difference Methods for a $d$-Dimensional Hypercube

Qiwei Feng, Bin Han, Michelle Michelle, Jiwoon Sim

TL;DR

This work develops high-order, compact, and symmetric finite difference methods for the variable-coefficient Poisson equation on a $d$-dimensional unit cube, producing symmetric positive definite systems on uniform grids. A 1D theory shows compact symmetric schemes can achieve arbitrary order, and the authors then extend to $d\ge 2$ by combining 1D and 2D directional stencils to obtain 4th-order schemes in general (with a special 2D case allowing 6th order under a derivative condition on $a$). They prove general upper bounds on the achievable order for compact symmetric FDMs in higher dimensions and provide explicit 4th-order (and some 6th-order in special cases) stencils, along with a dimension-reduction framework and reproducible numerical experiments validating the theory and solver performance. The approach yields stable, efficient solvers due to SPD systems and compact storage, and the explicit stencils facilitate straightforward implementation in practice.

Abstract

This paper presents compact, symmetric, and high-order finite difference methods (FDMs) for the variable Poisson equation on a $d$-dimensional hypercube. Our scheme produces a symmetric linear system: an important property that does not immediately hold for a high-order FDM. Since the model problem is coercive, the linear system is in fact symmetric positive definite, and consequently many fast solvers are applicable. Furthermore, the symmetry combined with the minimum support of the stencil keeps the storage requirement minimal. Theoretically speaking, we prove that a compact, symmetric 1D FDM on a uniform grid can achieve arbitrary consistency order. On the other hand, in the $d$-dimensional setting, where $d \ge 2$, the maximum consistency order that a compact, symmetric FDM on a uniform grid can achieve is 4. If $d=2$ and the diffusion coefficient satisfies a certain derivative condition, the maximum consistency order is 6. Moreover, the finite compact, symmetric, 4th-order FDMs for $d\ge 3$, can be conveniently expressed as a linear combination of two types of FDMs: one that depends on partial derivatives along one axis, and the other along two axes. All finite difference stencils are explicitly provided for ease of reproducibility.

High-order, Compact, and Symmetric Finite Difference Methods for a $d$-Dimensional Hypercube

TL;DR

This work develops high-order, compact, and symmetric finite difference methods for the variable-coefficient Poisson equation on a -dimensional unit cube, producing symmetric positive definite systems on uniform grids. A 1D theory shows compact symmetric schemes can achieve arbitrary order, and the authors then extend to by combining 1D and 2D directional stencils to obtain 4th-order schemes in general (with a special 2D case allowing 6th order under a derivative condition on ). They prove general upper bounds on the achievable order for compact symmetric FDMs in higher dimensions and provide explicit 4th-order (and some 6th-order in special cases) stencils, along with a dimension-reduction framework and reproducible numerical experiments validating the theory and solver performance. The approach yields stable, efficient solvers due to SPD systems and compact storage, and the explicit stencils facilitate straightforward implementation in practice.

Abstract

This paper presents compact, symmetric, and high-order finite difference methods (FDMs) for the variable Poisson equation on a -dimensional hypercube. Our scheme produces a symmetric linear system: an important property that does not immediately hold for a high-order FDM. Since the model problem is coercive, the linear system is in fact symmetric positive definite, and consequently many fast solvers are applicable. Furthermore, the symmetry combined with the minimum support of the stencil keeps the storage requirement minimal. Theoretically speaking, we prove that a compact, symmetric 1D FDM on a uniform grid can achieve arbitrary consistency order. On the other hand, in the -dimensional setting, where , the maximum consistency order that a compact, symmetric FDM on a uniform grid can achieve is 4. If and the diffusion coefficient satisfies a certain derivative condition, the maximum consistency order is 6. Moreover, the finite compact, symmetric, 4th-order FDMs for , can be conveniently expressed as a linear combination of two types of FDMs: one that depends on partial derivatives along one axis, and the other along two axes. All finite difference stencils are explicitly provided for ease of reproducibility.

Paper Structure

This paper contains 14 sections, 6 theorems, 74 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Organization
  4. Notations and definitions
  5. Construction of compact, symmetric, and arbitrary high-order 1D FDMs
  6. Compact, symmetric, and fourth-order $d$-dimensional FDMs, $d\geqslant 2$
  7. The 2D and 3D cases
  8. The general $d$-dimensional case
  9. Numerical experiments
  10. Derivations of compact, symmetric, and high-order FDMs in general dimensions
  11. Expansion of the solution
  12. Structure of the stencil coefficients
  13. Maximum consistency order of $d$-dimensional compact FDMs for $d\geqslant 2$
  14. Conclusion

Key Result

Theorem 3.1

Let $M$ be a positive integer, $0<h<1$, and $c^* \in (0,1)$ with $(c^* -h, c^* + h) \subset (0,1)$. Assume that $a$ and $f$ are functions that are $M$ and $M-1$ times differentiable respectively. There is a compact, symmetric, and $M$th-order consistent finite difference scheme for eq:PDE given by where $C_{-1}$, $C_0$, $C_1$, and $d_{\ell}$ are defined as such that and the quantities $E_{j,1}$

Figures (2)

  • Figure 1: \ref{['Example:2']}: The coefficient $a$ (left), the exact solution $u$ (middle), and the error $|u_h-u|$ (right) on $[0,1]^2$ with $h=2^{-10}$, where $u_h$ is obtained by using the compact, symmetric, and 4th-order 2D FDM.
  • Figure 2: \ref{['Example:3']}: The coefficient $a$ on $[-1,1]^3$ (top left), the exact solution $u$ on $[-1,1]^3$ (top right), the error $|u_h-u|$ on the subdomain $[-1,0]^3$ (bottom left), and the error $|u_h-u|$ on the subdomain $[-1/2,1/2]^3$ (bottom right) with $h=2/2^{7}$, where $u_h$ is obtained by using the compact, symmetric, and 4th-order 3D FDM.

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 3.1
  • Proof 1
  • Theorem 4.1
  • Proof 2
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • Lemma 6.1
  • Proof 3
  • ...and 6 more