A Simple Analytic Solution of the Discrete Navier-Stokes Equations

A Simple Analytic Solution of the Discrete Navier-Stokes Equations Bik Kuang Min Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), Bangi, Selangor, Malaysia Email: [email protected] ORCID: 0009-0009-8368-3858 16 October 2025 Abstract We present a simple analytic solution of the Discrete Navier–Stokes Equations (DNSE) formulated entirely in operator form, without the use of index notation. The mathematical foundation is established through the definition of discrete functions, shift operators, and discrete vector fields, leading to a compact and self-consistent formulation of the DNSE. As a case study, we derive an explicit analytic solution for the 2D discrete Poiseuille flow, demonstrating that the discrete operator approach naturally reproduces the structure and behavior of its continuous counterpart. This work shows that discrete field formulations can possess fully analytic solutions, offering a unified and elegant framework for future extensions to higher-dimensional and nonlinear systems. Keywords: Discrete Navier–Stokes equations, operator formalism, discrete vector field, Poiseuille flow, analytic solution. 1 Introduction The Navier–Stokes equations (NSE) [3]form the fundamental framework for modeling fluid motion in the continuum. However, when these equations are expressed in their discrete counterparts, they are often treated merely as numerical approximations of the continuous model rather than as independent mathematical systems. Classical finite-difference and finite-volume formulations aim to approximate derivatives on a grid, but they do not endow the discrete setting with a self-contained operator structure. In this work, we propose and analyze the Discrete Navier–Stokes Equations (DNSE), formulated entirely in operator form. Unlike traditional discretizations written with subscript indices (e.g., ui,j), the present formulation is built upon 1 the algebra of discrete operators—forward, backward, and central differences— acting directly on discrete vector fields. This framework preserves the intrinsic algebraic and differential properties of the continuous equations while remaining fully within the discrete domain. As a case study, we derive an analytic solution for the two-dimensional steady discrete Poiseuille flow. Remarkably, the resulting expression is structurally identical to the classical continuous solution, demonstrating that the discrete operator formalism is internally consistent and physically coherent without requiring any continuum limit. This result suggests that discrete operator systems may represent an independent mathematical foundation for fluid dynamics, in which continuum models arise as smooth limits of fundamentally discrete dynamics. The paper is organized as follows: Section 2 introduces the necessary mathematical definitions and notation; Section 3 formulates the discrete Navier–Stokes equations; Section 4 presents the analytic solution for the discrete Poiseuille flow; Section 5 and Section 6 concludes with perspectives and future directions. 2 Mathematical Foundations The following terminology, notation, and definitions are adapted from standard textbooks on partial differential equations and difference equations [1,2,4]. Definition 2.1 (Discrete Function).Adiscrete function is a mapping f:Ω⊆Zn→C, where Znis the discrete n-dimensional integer lattice, and Cis the codomain of the function (real, integer, or complex values depending on context). Definition 2.2 (Shift Operator).Let x:Z→C. We define the shift operator Ekacting on a discrete function x(t)by Ekx:= x(t+k) for any integer k∈Z. Definition 2.3 (Partial Shift Operator).Given a discrete scalar field u(x1, x2, . . . , xn), the partial shift operator Eki xiacts on uby shifting the i-th coordinate: Eki xiu:= u(x1, . . . , xi+ki, . . . , xn) for any integer ki∈Z. Definition 2.4 (Difference Operator).Let x:Z→C. We define the (forward) difference operator ∆by ∆x:= x(t+1)−x(t) We may define the shift operator Eas Ex := x(t+ 1), so that the difference operator can be written compactly as ∆x=Ex −x 2 Definition 2.5 (Partial Difference Operator).Let u:Zn→C, and denote u=u(x1, x2, . . . , xn). The partial difference operator with respect to the variable xiis defined as: ∆xiu:= Exiu−u where Exiis the partial shift operator acting on the i-th coordinate: Exiu:= u(x1, . . . , xi+ 1, . . . , xn) Definition 2.6 (Backward Difference Operator).Let x:Z→Cbe a discrete function with x=x(t). Then the backward difference operator is defined as δx := x(t)−x(t−1)=x−E−1x Definition 2.7 (Partial Backward Difference Operator).Let u:Zn→C, u =u(x1, x2, . . . , xn) Then the partial backward difference operator with respect to xiis defined as δxiu:= u−E−1 xiu where E−1 xiis the backward shift operator acting on the xi-th coordinate. Definition 2.8 (Central Difference Operator).Let x:Z→C, with x=x(t). Then the central difference operator is defined as ∆cx:= 1 2Ex −E−1x where Eis the forward shift operator: Ex =x(t+ 1), and E−1x=x(t−1). Definition 2.9 (Partial Central Difference Operator).Let u:Zn→C, where u=u(x1, x2, . . . , xn). Then the partial central difference operator with respect to xiis defined as ∆c xiu:= 1 2Exiu−E−1 xiu, where Exiis the shift operator in the xi-direction, defined by Exiu=u(x1, . . . , xi+ 1, . . . , xn). Definition 2.10 (Discrete Gradient).Let u:Zn→Cbe a discrete scalar function, where u=u(x1, x2, . . . , xn). Then the discrete gradient of uis defined as the vector: ∇u:= (∆x1u, ∆x2u, . . . , ∆xnu) where ∆xiis the forward partial difference operator defined by ∆xiu:= u(x1, . . . , xi+ 1, . . . , xn)−u(x1, . . . , xi, . . . , xn) for each i= 1,2, . . . , n. 3 Definition 2.11 (Backward Gradient).Let u:Zn→C, u =u(x1, x2, . . . , xn), then the backward gradient is defined as ∇δu= (δx1u, δx2u, . . . , δxnu), where each δxiudenotes the partial backward difference in the xi-direction. Definition 2.12 (Central Gradient).Let u:Zn→C, u =u(x1, x2, . . . , xn), be a discrete function on the n-dimensional integer lattice. Then the Central Gradient is defined as the vector ∇cu:= ∆c x1u, ∆c x2u, . . . , ∆c xnu, where each ∆c xiudenotes the partial central difference of uwith respect to the variable xi. Definition 2.13 (Discrete Laplacian).Let u:Zn→C, u =u(x1, x2, . . . , xn), then the Discrete Laplacian is defined as ∇2u=∇δ· ∇u= n X i=1 δxi∆xiu, where ∇u= (∆x1u, . . . , ∆xnu)is the discrete gradient, and ∇δ= (δx1, . . . , δxn) is the backward gradient. Definition 2.14 (Ordinary Difference Equation of Order k).An ordinary difference equation of order kis an equation involving a single-variable function x:Z→C, expressed in terms of shift operators: F(Enx, En−1x, . . . , Ex, x, E−1x, . . . , E−mx, t) = 0, where Eix:= x(t+i),n, m ∈Z+,k=m+nis the total order, and Fis a (possibly nonlinear) function. [2] Definition 2.15 (Linear Ordinary Difference Equation).Let x:Z→C, x = (t), be a discrete function. A Linear Ordinary Difference Equation (O∆E) is an equation of the form n X k=m ak(t)Ekx=f(t), where m, n ∈Z, the coefficients ak:Z→Care given functions, and the shift operator Ekis defined by Ekx:= x(t+k). 4 Definition 2.16 (Partial Difference Equation).Let u:Ω⊆Zn→R(or C) be a scalar function defined on the discrete lattice. A partial difference equation (P∆E) with constant order is an equation of the form: FEk1 x1Ek2 x2···Ekn xnu(k1,k2,...,kn)∈S, x1, x2, ..., xn= 0, where: •u=u(x1, x2, ..., xn) •Fis a given function, which may be linear or nonlinear; •Exidenotes the shift operator in the xidirection, defined by Eki xiu=u(x1, x2, . . . , xi+ki, . . . , xn); •S⊂Znis a finite index set determining the set of applied shifts. Definition 2.17 (Discrete Vector Field).Adiscrete vector field on a discrete domain Ω⊆Znis a mapping F: Ω →Rm, that assigns to each discrete point x∈Ωa vector F(x)=(F1(x), F2(x), . . . , Fm(x)). 3 Discrete Navier–Stokes Equations We now propose the Discrete Navier–Stokes Equations (DNSE), written entirely in operator form: ρ∆tu+u· ∇cu=−∇cp+µ∇2u+f,∇c·u= 0, where u= (u, v, w) = (u(t, x, y, z), v(t, x, y, z), w(t, x, y, z)), t ∈N0,(x, y, z)∈Z3. Here, uisadiscrete vector field, u:N0×Z3→R3, and the operators ∆t,∇c, and ∇2denote the forward difference, central gradient, and discrete Laplacian, respectively. This system constitutes a set of nonlinear, non-autonomous partial difference equations. Relation to the continuous Navier–Stokes equations. Recall that in the continuous case, ρ∂u ∂t +u· ∇u=−∇p+µ∇2u+f,∇ · u= 0. 5 Each differential operator can be expressed as the limit of its corresponding finite-difference operator: ∂u ∂t = lim ht→0 u(t+ht, x)−u(t, x) ht ,∂u ∂x = lim hx→0 u(t, x +hx)−u(t, x −hx) 2hx , ∂2u ∂x2= lim hx→0 u(t, x +hx)+u(t, x −hx)−2u(t, x) h2 x . Setting the step sizes ht=hx=hy=hz= 1 recovers the discrete operator formulation given above. Thus, the DNSE can be viewed as a self-contained discrete analogue of the continuous Navier–Stokes equations, rather than a mere numerical approximation. 4 2D Discrete Poiseuille Flow We consider the two-dimensional autonomous form of the discrete Navier–Stokes equations: ∆tu+u∆c xu+v∆c yu=−1 ρ∆c xp+η(δx∆xu+δy∆yu),(1) ∆tv+u∆c xv+v∆c yv=−1 ρ∆c yp+η(δx∆xv+δy∆yv),(2) ∆c xu+ ∆c yv= 0.(3) Here u=u(t, x, y), v =v(t, x, y), t ∈N0,(x, y)∈Z2, and ρ,ηdenote the density and dynamic viscosity, respectively. 4.1 Simplifying assumptions To obtain an analytically tractable form, we impose the following restrictions: ∆tu= ∆tv= 0, u =u(y), v = 0, p =p(x), together with the boundary conditions u(t, x, 0) = 0, u(t, x, h) = 0. Under these assumptions, equations (1)–(2) reduce to a single scalar equation: η δy∆yu=1 ρ∆c xp. (4) Dividing by ηgives δy∆yu=1 ρη ∆c xp. (5) 6 4.2 Analytic solution Let q=q(x)=∆c xp. Then equation (5) can be written in shift-operator form as (1 −E−1 y)(Ey−1)u=q ρη ,(6) or equivalently, Eyu+E−1 yu−2u=q ρη .(7) Equation (7) is a non-homogeneous linear ordinary difference equation with constant coefficients. Homogeneous solution. The associated homogeneous equation Eyu+E−1 yu−2u= 0 has the characteristic equation r+r−1−2=0 ⇒(r−1)2= 0, yielding the homogeneous solution uh(y) = A+By. Particular solution. For the particular solution, we assume up(y)=Cy2. Substitution into (7) gives Eyup+E−1 yup−2up= 2C=q ρη , so that C=q 2ρη . Hence the general solution is u(y) = A+By +q 2ρη y2. Boundary conditions. Applying the boundary conditions u(0) = 0 and u(h) = 0 yields A= 0, B =−q 2ρη h. Therefore, the velocity profile is u(y) = q 2ρη y(y−h).(8) 7 Final form. Recalling that q= ∆c xp, the steady-state discrete Poiseuille solution is u(t, x, y) = 1 2ρη ∆c xp y(y−h),(9) v(t, x, y)=0.(10) 4.3 Discussion The result shows that the discrete Navier–Stokes equations reproduce the same parabolic velocity profile as the continuous Poiseuille flow, without invoking any limiting process or approximation. This demonstrates that the discrete formulation is self-consistent and physically complete, possessing analytic structure equivalent to its continuous counterpart. 5 Conclusion The present study demonstrates that the discrete Navier–Stokes equations, formulated in terms of shift and difference operators, yield a mathematically exact and physically consistent description of two-dimensional viscous flow. Without invoking any continuum limit or numerical approximation, the discrete operator algebra naturally produces the parabolic velocity profile characteristic of Poiseuille flow. This finding establishes that the discrete formulation preserves both conservation laws and boundary conditions inherent to classical fluid mechanics. Hence, the discrete Navier–Stokes system represents a self-contained and physically complete framework, rather than a numerical surrogate of the continuous equations. More broadly, this result points toward a deeper philosophical view of physical law: the continuous equations of fluid dynamics may be regarded as smooth limits of a more fundamental discrete dynamics. The discrete model thus provides a unified mathematical and physical foundation from which the continuum description emerges as an approximation. 6 Future Work The present study demonstrates that the discrete operator formulation of the Navier–Stokes equations admits fully analytic solutions in steady-state conditions. Future work will extend this framework to: •Non-steady and time-dependent discrete flows, including transient evolution and boundary-driven instabilities; •Higher-dimensional discrete vector fields, particularly 3D discrete vortex dynamics and rotational flows; 8 •Nonlinear coupling between discrete momentum and pressure fields, exploring the onset of discrete turbulence; •Connections between discrete operator algebras and lattice-based statistical mechanics models, such as the Ising model and the Sandpile model. These directions aim to establish a comprehensive theory of discrete fluid dynamics, where continuum equations arise as smooth limits of an intrinsically discrete operator framework. Acknowledgement The author confirms that all models, equations, and core results are original. Generative AI tools (ChatGPT, GPT-5, OpenAI) were used only for LaTeX formatting, language polishing, and rephrasing of well-known theorems; all mathematical ideas, proofs, and formulations are the author’s own responsibility. References [1] Kuang Min Bik. On the theory of partial difference equations: From numerical methods to language of complexity. Preprints, 2025. Preprint, version 2, posted 13 August 2025, non peer-reviewed. [2] Saber Elaydi. An Introduction to Difference Equations. Undergraduate Texts in Mathematics. Springer, New York, 3rd edition, 2005. [3] G. P. Galdi. An Introduction to the Mathematical Theory of the Navier– Stokes Equations: Steady-State Problems. Springer Monographs in Mathematics. Springer, New York, 2011. Second edition. [4] Peter J. Olver. Introduction to Partial Differential Equations. Undergraduate Texts in Mathematics. Springer, Cham, Heidelberg, New York, Dordrecht, London, 2014. 9