Discrete-Time Euler Equation

• Discrete-Time Euler Equation is a discrete analogue of the classical Euler ODE, defined through finite-difference methods that preserve analytical solution classes.
• The finite-operator approach ensures consistency and stability by recovering the continuous form as the discretization parameter tends to zero, enhancing numerical ODE integration.
• Extensions to fractional calculus and applications in optimal transport and fluid dynamics showcase its versatility in addressing complex variational and discrete-analytical problems.

The discrete-time Euler equation refers to a broad family of recurrence and difference equations, both as direct discretizations of the classical Cauchy–Euler ordinary differential equation (ODE) and as finite-difference manifestations of variational principles in optimization and physics. These discrete analogues are essential in numerical analysis, variational calculus on time scales, fractional calculus, and, more recently, multi-marginal optimal transport and fluid dynamics.

1. Discrete Analogues of the Cauchy–Euler ODE

The Cauchy–Euler ODE,

$x^2 u''(x) + a x u'(x) + b u(x) = 0,$

admits a canonical discretization on a uniform lattice $x_n = nh$ ($h > 0$). The delta operator $\Delta$ replaces $d/dx$, acting as

$$\Delta u_n = \frac{u_{n+1} - u_n}{h}, \qquad \Delta^2 u_n = \frac{u_{n+2} - 2u_{n+1} + u_n}{h^2}.$$

Expanding $u(x)$ in a basis of factorial (basic) polynomials,

$$(x)_k = \prod_{j=0}^{k-1}(x - jh), \quad u(x) = \sum_{k=0}^{\infty} \zeta_k (x)_k,$$

yields the lattice representation

$u_n = \sum_{k=0}^n \zeta_k h^k \frac{n!}{(n-k)!}.$

Rewriting the ODE using the finite Rota/umbral calculus, one derives the three-term recurrence,

$$\bigl(n^2 + (a-1)n + b\bigr) u_n - n(a + 2n - 2) u_{n-1} + n(n - 1) u_{n-2} = 0, \qquad n \geq 2,$$

with boundary conditions given by $u_0$ and $u_1$. This approach algorithmically inherits classes of exact solutions $u(x) = x^r$ as discrete analogues $u_n = (nh)_r = \frac{n!}{(n - r)!} h^r$, connected to roots of the indicial equation $r(r - 1) + a r + b = 0$ (Rodríguez et al., 7 Jul 2025).

2. Structure and Properties of the Discretized Equation

This finite-operator discretization exploits the Rota algebra structure, where delta operators act on a nonlocal $*_Q$-product with a genuine Leibniz rule: $Q(f *_Q g) = (Qf) *_Q g + f *_Q (Qg).$ The position operator $\beta$ acts as $[Q, x\beta] = 1$, generating basic polynomials $p_n(x) = (x\beta)^n \cdot 1$, and ensuring integrability and structural properties faithful to the continuous case.

The discrete model maintains stability and consistency:

• Consistency: As $h \to 0$ and $n \to \infty$ with $x = nh$ fixed, the recurrence recovers the differential form of the continuous Euler ODE.
• Stability: The presence of exact solutions for modes corresponding to the roots of the indicial equation suggests boundedness, though no von Neumann or CFL-type analysis is presented.

3. Classical Discrete-Time Euler Updates in ODE Integration

The standard (first-order) discrete-time Euler method for an initial value problem $y'(t) = f(y(t))$, $y(0) = 0$, on a partition $Q = (q_0, ..., q_k)$ is given by

$$y_{j+1} = y_j + \Delta_j f(y_j), \quad \Delta_j = q_{j+1} - q_j.$$

A second-order Euler operator $E_2$ defined using an interval extension of $f$ and its derivative achieves higher-order accuracy: $$y(q_{j+1}) = y(q_j) + \Delta_j u(A_j) + \frac{1}{2} \Delta_j^2 (u' \cdot u)(A_j),$$ where $A_j = y(q_j) + M \Delta_j$, $M$ is a Lipschitz constant for $f$. The second-order operator exhibits $O(h^2)$ global error, outperforming both first-order Euler and Runge–Kutta Euler methods in accuracy and computational efficiency for Lipschitz $f$ (Edalat et al., 2023).

The computability of these discrete-time Euler operators is formulated in continuous domains of left-continuous interval-valued maps, with implementations based on arbitrary-precision interval arithmetic.

4. The Discrete Euler–Lagrange Equation on Time Scales

On uniform or arbitrary time scales $\mathbb{T}$, discrete variational calculus leads to Euler–Lagrange-type recurrence relations. For a grid $\mathbb{Z}$ or $h\mathbb{Z}$ with delta operator $\Delta x_k = x_{k+1} - x_k$, the discrete Euler–Lagrange equation for a functional $I(x) = \sum_{k=0}^{N-1} L(k, x_k, \Delta x_k)$ is

$$\left. \frac{\partial L}{\partial x_k} \right|_{(x_k, \Delta x_k)} - \left[ \frac{\partial L}{\partial (\Delta x_k)}\Big|_{(x_k, \Delta x_k)} - \frac{\partial L}{\partial (\Delta x_{k-1})}\Big|_{(x_{k-1}, \Delta x_{k-1})} \right] = 0.$$

This discrete equation is the direct analogue of the continuous Euler–Lagrange equation in the calculus of variations and is derived using a discrete integration by parts formula. The necessary self-adjointness condition and the equation of variation ensure the variational structure is preserved in the discrete setting (Dryl et al., 2014).

5. Fractional and Generalized Discrete-Time Euler Equations

The discrete-time Euler equation admits generalizations to fractional orders via discrete-time fractional calculus. On $h\mathbb{Z}$ with graininess $\mu(t) = h$, fractional forward and backward difference operators ${}_a \Delta_h^\alpha$ and ${}_h \Delta_b^\alpha$ are defined by compositions of fractional $h$-sums and delta derivatives.

For a variational problem,

$$\mathcal{L}(y) = \int_a^b L(t, y^\sigma(t), {}_a \Delta_h^\alpha y(t), {}_h \Delta_b^\beta y(t))\,\Delta t,$$

the fractional discrete Euler–Lagrange equation is

$$L_u[\hat y](t) + {}_h\Delta_{\rho(b)}^\alpha(L_v[\hat y])(t) + {}_a\Delta_h^\beta(L_w[\hat y])(t) = 0,$$

where fractional $h$-summation by parts is used to transfer fractional differences in the variational derivative (Bastos et al., 2010).

As $h \to 0$, the discrete-fractional equation recovers the continuous-time Riemann–Liouville fractional Euler–Lagrange equation, and, when $\alpha, \beta$ are integers, it reduces to the classical discrete equation.

6. Discrete-Time Euler Equations in Fluid Dynamics and Optimal Transport

The discrete-time Euler equations also emerge as the central equations in the time-discrete variational formulations of incompressible fluid mechanics and multi-marginal optimal transport (MMOT). Starting from Arnold's variational principle for incompressible flow, the Euler–Lagrange equations are recast, upon discretization, as minimization problems over probability measures on path space:

$$\min_{\gamma \in \mathcal{P}(\Omega^{N+1})} \int_{\Omega^{N+1}} \left( \sum_{i=1}^N \frac{|\omega_i - \omega_{i-1}|^2}{t_i - t_{i-1}} \right) d\gamma(\omega),$$

subject to appropriate marginal and endpoint constraints.

A fundamental phenomenon in these discrete-time equations is the occurrence of mass-splitting: for $N \geq 3$, optimal couplings $\gamma$ are generally not of Monge form; that is, they do not correspond to deterministic maps at intermediate time steps. Explicit examples demonstrate non-Monge solutions both on finite grids and in continuous one-dimensional domains, signifying that such mass-splitting is an inherent feature of the time-discrete Euler optimal transport cost (Friesecke, 6 Jan 2026).

7. Comparative Analysis and Mathematical Impact

The Galois/finite-operator discretization of the Cauchy–Euler equation fundamentally differs from classical one-step discrete-time methods. While the latter do not preserve analytical solution classes of their continuous counterparts, the finite-operator approach is designed to inherit exact solutions, restore the genuine Leibniz property, and respect algebraic structures such as the Heisenberg–Weyl algebra on the lattice (Rodríguez et al., 7 Jul 2025). Fractional and variational generalizations extend these principles, enabling discrete analysis methods consistent with both classical and fractional calculus of variations.

In numerical practice, higher-order Euler discretizations outperform first-order and some Runge–Kutta methods under weaker regularity assumptions (Edalat et al., 2023). In the context of optimal transport and fluid equations, the failure of Monge form in the time-discrete Euler problem highlights a deep structural shift from continuous to discrete settings, with mass-splitting and Kantorovich-type solutions becoming generic as soon as three or more time marginals are considered (Friesecke, 6 Jan 2026).

These advances collectively anchor the discrete-time Euler equation as a central object across numerical ODE theory, calculus of variations on discrete structures, fractional dynamics, and modern optimal transport.