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 $x_n = nh$0 and $x_n = nh$1. This approach algorithmically inherits classes of exact solutions $x_n = nh$2 as discrete analogues $x_n = nh$3, connected to roots of the indicial equation $x_n = nh$4 (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 $x_n = nh$5-product with a genuine Leibniz rule: $x_n = nh$6 The position operator $x_n = nh$7 acts as $x_n = nh$8, generating basic polynomials $x_n = nh$9, and ensuring integrability and structural properties faithful to the continuous case.

The discrete model maintains stability and consistency:

• Consistency: As $h > 0$0 and $h > 0$1 with $h > 0$2 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 $h > 0$3, $h > 0$4, on a partition $h > 0$5 is given by

$h > 0$6

A second-order Euler operator $h > 0$7 defined using an interval extension of $h > 0$8 and its derivative achieves higher-order accuracy: $h > 0$9 where $\Delta$0, $\Delta$1 is a Lipschitz constant for $\Delta$2. The second-order operator exhibits $\Delta$3 global error, outperforming both first-order Euler and Runge–Kutta Euler methods in accuracy and computational efficiency for Lipschitz $\Delta$4 (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 $\Delta$5, discrete variational calculus leads to Euler–Lagrange-type recurrence relations. For a grid $\Delta$6 or $\Delta$7 with delta operator $\Delta$8, the discrete Euler–Lagrange equation for a functional $\Delta$9 is

$d/dx$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 $d/dx$1 with graininess $d/dx$2, fractional forward and backward difference operators $d/dx$3 and $d/dx$4 are defined by compositions of fractional $d/dx$5-sums and delta derivatives.

For a variational problem,

$d/dx$6

the fractional discrete Euler–Lagrange equation is

$d/dx$7

where fractional $d/dx$8-summation by parts is used to transfer fractional differences in the variational derivative (Bastos et al., 2010).

As $d/dx$9, the discrete-fractional equation recovers the continuous-time Riemann–Liouville fractional Euler–Lagrange equation, and, when $$\Delta u_n = \frac{u_{n+1} - u_n}{h}, \qquad \Delta^2 u_n = \frac{u_{n+2} - 2u_{n+1} + u_n}{h^2}.$$0 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:

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

subject to appropriate marginal and endpoint constraints.

A fundamental phenomenon in these discrete-time equations is the occurrence of mass-splitting: for $$\Delta u_n = \frac{u_{n+1} - u_n}{h}, \qquad \Delta^2 u_n = \frac{u_{n+2} - 2u_{n+1} + u_n}{h^2}.$$2, optimal couplings $$\Delta u_n = \frac{u_{n+1} - u_n}{h}, \qquad \Delta^2 u_n = \frac{u_{n+2} - 2u_{n+1} + u_n}{h^2}.$$3 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.