Spatial discretization of partial differential equations with integrals

TL;DR

This work develops a systematic finite-difference framework to discretize PDEs while preserving prescribed integrals by constructing skew-symmetric (p+1)-tensors $K$ on a grid. It unifies Hamiltonian and non-Hamiltonian integral preservation through a discrete operator formalism, enabling energy, enstrophy, and Casimir-like conservation via $\dot u = K(\nabla I^1,\nabla I^2,\dots)$. The paper provides a constructive methodology, including a diagrammatic approach and symmetry-guided averaging, to build $K$ that approximates continuous operators and yields Arakawa-type Jacobians in 2D, with adaptations to various grids (square, triangular, FCC) and higher dimensions. These integral-preserving discretizations offer improved nonlinear stability and fidelity of long-time dynamics, with practical implications for accurately simulating conservative PDEs on arbitrary grids.

Abstract

We consider the problem of constructing spatial finite difference approximations on a fixed, arbitrary grid, which have analogues of any number of integrals of the partial differential equation and of some of its symmetries. A basis for the space of of such difference operators is constructed; most cases of interest involve a single such basis element. (The ``Arakawa'' Jacobian is such an element.) We show how the topology of the grid affects the complexity of the operators.

Figures (5)

• Figure 1: Case 2, Two integrals, one space dimension. (a) One variable; (b) $m$ variables. See Eq. (\ref{['p2d1']}) for the finite difference interpretation of (a).
• Figure 2: Case 2, Two integrals, one space dimension, $m$ variables, ${\hbox{\boldmath$l$\unboldmath}}=(0,0,1)$.
• Figure 3: Case 4, Step-by-step construction of the Arakawa Jacobian.
• Figure 4: Case 4, The Arakawa Jacobian on a triangular grid.
• Figure 5: Case 5, 3 integrals in 2 dimensions