Conservation Law of Discrete Gradient Integrators

• The Conservation Law of GDI is a framework that formalizes the preservation of energy-like invariants in numerical discretizations of Hamiltonian PDEs.
• Discrete Gradient Integrators replace continuous derivatives with discrete gradients, enabling time-discrete analogues that maintain key invariants at each step.
• Its implementation in methods such as the AVF scheme and applications like the KdV equation demonstrates practical benefits for long-term simulation stability.

The conservation law of Discrete Gradient Integrators (GDI) formalizes the preservation of an energy-like quantity in numerical discretizations of Hamiltonian partial differential equations (PDEs). GDI methods, based on discrete gradients, yield time-discrete conservation laws that mirror the structure of their continuous analogues. This principle is foundational for the numerical integration of conservative systems, ensuring the preservation of key invariants such as energy at the discrete level (McLachlan et al., 2013).

1. Continuous Conservation Laws in Hamiltonian Systems

For a Hamiltonian PDE of the form

$z_t = K\,\delta H/\delta z\,,$

with $z(x,t) \in \mathbb{R}^n$, constant antisymmetric $K$, and a Hamiltonian $H[z] = \int E(z, z_x, z_{xx}, \ldots)\,dx$, the energy density $E$ satisfies the local conservation law

$$\partial_t\,E(z, z_x, \ldots) + \partial_x\,F(z, z_x, \ldots) = 0\,,$$

where the flux $F$ is a local quadratic form in the variational derivatives of $E$. In canonical forms, $F = E_{z_x}^\top K E_z$. This expresses the essential property that the time rate of change of the conserved density is balanced by the flux divergence.

2. Discrete Gradient: Definition and Properties

A discrete gradient $\nabla\bar{E}: V \times V \rightarrow V$ for any differentiable $E: V \to \mathbb{R}$ satisfies

$$\langle z^{n+1} - z^n,\, \nabla\bar{E}(z^n, z^{n+1}) \rangle = E(z^{n+1}) - E(z^n)\,,$$

with the consistency condition $\nabla\bar{E}(z, z) = \nabla E(z)$. For example, the Average Vector Field (AVF) discrete gradient is

$$\nabla\bar{E}(z^n, z^{n+1}) = \int_0^1 \nabla E\big(\theta z^{n+1} + (1-\theta) z^n \big)\, d\theta\,.$$

These discrete gradients replace the role of standard derivatives in constructing time-discrete analogues of conservation laws, preserving the relevant invariants exactly at each time step.

3. Discrete Gradient Integrators and the Energy Conservation Law

Applied to a Hamiltonian PDE, the GDI time-discrete scheme is

$$\frac{z^{n+1} - z^n}{\Delta t} = K\, \nabla\bar{H}(z^n, z^{n+1})\,,$$

with $K$ constant antisymmetric. The central theorem is that the same conserved density $E$ satisfies a time-discrete conservation law:

$$E(z^{n+1}, z^{n+1}_x, ...) - E(z^n, z^n_x, ...) + \Delta t\, \partial_x F^d(z^n, z^{n+1}) = 0\,,$$

where the discrete flux $F^d$ is obtained by replacing each variational derivative $\partial E/ \partial (\cdot)$ in the continuous $F$ by its discrete gradient component:

$$F^d = \left( \nabla\bar{E}_{z_x} \right)^\top K \nabla\bar{E}_{z}\,.$$

In the fully discrete (space–time) setting, the spatial derivative is replaced by a suitable finite-difference (summation-by-parts) operator.

4. Theoretical Basis and Proof Mechanism

The conservation law follows from multiplying the discrete step by the discrete gradient and employing its defining axiom:

$$\frac{1}{\Delta t} \langle z^{n+1} - z^n, \nabla\bar{E} \rangle = \frac{E(z^{n+1}) - E(z^n)}{\Delta t} = \langle \nabla\bar{E}, K \nabla\bar{E} \rangle = 0\,,$$

by antisymmetry of $K$. The flux divergence structure is retained because discrete gradients replace the partial derivatives in the original continuous flux, and product rules hold at the discrete level analogously.

5. Canonical Formulas and Implementation

Key canonical formulas are:

Quantity	Continuous Case	Discrete Case
Conservation law	$\partial_t E + \partial_x F = 0$	$[E^{n+1}-E^n]/\Delta t + \partial_x F^d=0$
Flux	$F = E_{z_x}^\top K E_z$	$F^d = (\nabla\bar{E}_{z_x})^\top K \nabla\bar{E}_z$
Discrete gradient axiom	$\langle \Delta z, \nabla E \rangle = \Delta E$	$$\langle z^{n+1} - z^n, \nabla\bar{E} \rangle = E^{n+1} - E^n$$

These formulas enable the construction of algorithms in which energy (or other invariants) is preserved at the level of the difference scheme.

6. Illustrative Examples

Nonlinear Wave Equation

For

$q_t = p\,,\quad p_t = q_{xx} - V'(q)$

with energy density $E = \frac{1}{2}p^2 + \frac{1}{2} q_x^2 + V(q)$, the discrete GDI scheme preserves

$$\frac{E(q^{n+1}, p^{n+1}) - E(q^n, p^n)}{\Delta t} + D_x \left[ -\frac{p^{n+1}+p^n}{2} \frac{q^{n+1}+q^n}{2} \right]=0\,.$$

Korteweg–de Vries (KdV) Equation

For $u_t + 6u u_x + u_{xxx} = 0$ with Hamiltonian $H[u] = \int (\frac{1}{2}u_x^2 - u^3)\,dx$, a discrete-gradient step yields

$$\frac{H[u^{n+1}] - H[u^n]}{\Delta t} + D_x F^d = 0\,,$$

where $F^d$ replaces all derivatives in the continuous $F$ by discrete gradient expressions.

7. Significance and Scope

Any PDE system with constant $K$ Hamiltonian structure, discretized in time by a GDI, inherits a local time-discrete conservation law with the same energy density as the continuous system. The discrete flux is obtained by a systematic replacement of continuous derivatives by discrete gradients in the original flux. This property ensures energy preservation at the discrete level, which is crucial for the long-term stability and fidelity of numerical simulations of conservative systems. The conservation law is equally valid for space–time fully discrete schemes when summation-by-parts analogues of differential operators are used (McLachlan et al., 2013).