A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

Abstract

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately.

1. Introduction

Navier–Stokes equations, developed based on momentum and mass conservation principles, model Newtonian fluid flow dynamics successfully [1]. The resulting equations have been widely employed in fields such as aerodynamics, fluid mechanics, astrophysics, and weather forecasting [2,3]. In the majority of practical applications, Navier–Stokes equations can hardly be solved by analytical means, which, in practice, implies that resolution requires advanced numerical methods with high-performance computing [4]. Despite their wide usage, various extensions of Navier–Stokes equations, termed anomalous Navier–Stokes equations, have been proposed to model anomalous flow processes. Anomalous diffusion describes transport processes in which particles deviate from classical diffusion processes, and the mean square displacement no longer depends linearly on time. Still, it is instead related to a power-law dependence. Precisely, for an exponent that is lower than one, the process is slower and is called subdiffusion, while an exponent that is greater than one corresponds to a faster diffusion process, called superdiffusion [5].

One of the most important extensions to Navier–Stokes equations is the inclusion of fractional derivatives that allow a deeper characterization of fluid behavior under anomalous diffusion conditions [6]. The usual integer-order differential equations, together with their new counterparts, that is, fractional orders in differential equations, bring in a new class of differential equations known as fractional differential equations (FDEs), which have shown exceptional efficiency in modeling complex phenomena in physical and engineering processes that exhibit memory, long-range dependence, and non-local interactions [7]. Indeed, using fractional derivatives, it is possible to introduce memory effects and power-law relations that allow depicting diffusion phenomena more realistically [8,9,10]. They have been applied quite effectively to describe anomalous diffusion processes in which the mean square displacement departs from the linear growth characteristics of classical diffusion [5,11].

By systematically studying the properties and behavior of fractional Navier–Stokes equations (FNSE), much work has been carried out: the study of the well-posedness of mild solutions in Besov spaces [12], the decay properties analysis of weak solutions by using the Fourier splitting method [13], and the extension of Galdi’s classical results on energy equality for distributional solutions [14]. Others have investigated the approximate controllability of mild solutions with delay [15], introduced an approximate analytical solution via a generalized Laplace residual power series method [16], and proposed a reliable algorithm using the homotopy perturbation transform method [17]. The controlled Picard approach combined with the Laplace transform will be applied to the multi-dimensional time-FNSE driven from incompressible fluid flow, and we will propose an efficient method that can solve nonlinear fractional differential types without using Lagrange multipliers or Adomian polynomials [18]. Other results include the determination of critical values in fractional three-dimensional Navier–Stokes equations [19], an investigation on averaging between two-dimensional FNSE with singularly oscillating external forces [20], and research into the existence of uniform analytic solutions of the Cauchy problem in the critical Fourier–Herz spaces [21].

The numerical solution of FDEs is different from classical integer-order differential equations due to the type of problem. Unlike the solutions of classical differential equations, which are local to function shapes and information, solving FDEs requires a buffer that stores the solution history to obtain a horizon solution. This is because the fractional derivatives are non-local and dependent on memory; historical information must be considered to model the system’s behavior accurately. Several works have been proposed for efficiently solving FDEs and coping with their intensive computational needs, which usually involves a trade-off between accuracy and required memory [22,23]. The most popular numerical approaches, so far, to solve the system of FNSE are finite difference method [24,25], homotopy-based approaches [26,27], Laplace transform [28], Adomian decomposition method [29], and fractional-step methods [30,31]. Each has clear advantages in tackling fractional derivatives, nonlinearities, and multidimensional problems. The choice of method largely depends on problem formulation, the desired accuracy, and computational resources. Despite the diversity within this family of methods, they all share one basic principle: the solution representation at discrete grid points and approximations of derivatives are shown as the differences between these values. Thus, the solutions obtained using such methods result in expensive computational costs, besides difficulties in maintaining accuracy near boundaries. Moreover, their inability to efficiently handle complex, spatially intricate boundary geometries—where grid points must conform to intricate boundaries—presents a significant drawback.

In the recent couple of years, there has been an increasing interest in seeking alternative methods, for example, mesh-free approaches to solve spatial fractional partial differential equations (PDEs), since they do not require any predefined mesh structure; hence, these types of methods are suitable for problems involving complicated geometries or time-evolving domains [24]. The so-called mesh-free methods, representing radial basis function (RBF) methods [32], radial point interpolation (RPI) [25], moving least squares (MLS) [33], and the meshless local Petrov–Galerkin (MLPG) method [34], are themselves promising alternatives. The advantages of such methods include the flexibility in handling complicated geometries and avoiding the limitations arising in the case of mesh-based methods.

Meshless methods solve PDEs in strong and weak forms by applying arbitrarily distributed collocations in the solution domain [35]. These points are approximated by assumed global or local basis functions. Weak form methods create weak governing equations on local subdomains, allowing task decoupling and trial and test space selection. Most of the weak-based mesh-free methods suffer from the cost of numerical integrations. For instance, the MLPG method, as a weak form method, utilizes a local symmetric weak form and shape functions from the MLS approximation to solve boundary value problems and partial differential equations effectively [30]. The MLPG method does not require any background integration cells, and all the integrations are carried out locally over the small quadrature domains of regular shapes. Hence, many numerical integrations need to be computed over the complicated MLS shape functions, resulting in an expensive numerical scheme [30]. In contrast, strong form methods are meshless, straightforward, and have good convergence when using simple algorithms and coding. For instance, the RPI method avoids expensive integration by using a smart modification for which more accurate ones have replaced shape functions. The RPI method has successfully solved heat conduction [36,37], two and three-dimensional potential, and anisotropic diffusion problems [38].

This paper focuses primarily on developing a cost-effective and efficient mesh-free numerical methodology for two-dimensional FNSE problems in a conservation form, incorporating the stream function, vorticity, and advective terms used to model flow in a cavity [5,11]. That is,

$$\begin{array}{ccc}\hfill D^{\alpha }\mathbf{u}(t;\mathbf{x})& =& -\nabla p(t;\mathbf{x})+{\displaystyle \frac{1}{\mathrm{Re}}}{\nabla }^2\mathbf{u}(t;\mathbf{x})+\mathbf{f}\left(t\right),\hfill \\ \hfill \nabla \mathbf{u}(t;\mathbf{x})& =& 0,\hfill \end{array}$$

(1)

where $\mathbf{u}(t;\mathbf{x})$ is the flow velocity in a two-dimensional space ${\mathbf{x}}^{\top }=[x,y]$, $p(t;\mathbf{x})$ is the pressure, the Reynolds number is denoted by “Re”, the function $\mathbf{f}$ is a source vector, divergence is denoted by ∇, and $D^{\alpha }$ is called fractional material derivative, which is defined as  

$${\mathcal{D}}_t^{\alpha }(·)+\mathbf{u}(t;\mathbf{x})·\nabla (·)$$

in which ${\mathcal{D}}_t^{\alpha }u$ is the Caputo fractional derivative of order $\alpha$, $0<\alpha <1$, defined by

$${\mathcal{D}}_t^{\alpha }u={\displaystyle \frac{1}{\Gamma (1-\alpha )}}\underset{0}{\overset{T}{\int }}{\displaystyle \frac{u^{\prime }\left(\tau \right)}{{(t-\tau )}^{\alpha }}}d\tau ,$$

(2)

where $\Gamma (·)$ is the gamma function.

The results obtained in this investigation discuss the challenges of the accurate solution of the FNSE through a direct comparison with the analytical solution of classical Navier–Stokes equations, particularly for a fractional order close to 1. Numerical tests are performed for the benchmark problem of a Taylor–Green vortex, able to underline the performance and accuracy of such a method with both strengths and weaknesses. These parameters have been found to strongly interact with each other in a complicated way to provide the accuracy of the solution. The computations of the pressure field are of particular concern. In such a case, the present work emphasizes the importance of selecting the parameters for better accuracy and efficiency of simulations within computational fluid dynamics.

The scheme of this paper is as follows: Section 2 describes all accompanying definitions and theories that are used later on; the RPI shape functions are also introduced in this section, where the approximation of the time-fractional derivative is approached by the usual finite difference method. The proposed RPI method is developed in Section 3. In contrast, the efficiency of the proposed method is demonstrated for two illustrative problems in Section 4. Finally, in Section 5, conclusions of this present study are drawn.

2. Basic Definitions for Radial Point Interpolation

This section introduces the fundamental definitions and concepts that form the basis for the subsequent development and analysis. These preliminaries include key mathematical formulations and theoretical underpinnings necessary for understanding the proposed method.

Radial basis functions are a key component of the RPI method, providing a flexible and powerful means for interpolating scalar fields across a domain [39,40]. The RPI of a scalar function $u\left(\mathbf{x}\right):{\mathbb{R}}^d\mapsto \mathbb{R}$ in the domain $\Omega \subset {\mathbb{R}}^d$ at n points is given by

$$u\left(\mathbf{x}\right)=\sum _{i=1}^N{R}_i\left(\mathbf{x}\right)a_i+\sum _{j=1}^m{p}_j\left(\mathbf{x}\right)b_j=\{R{\left(\mathbf{x}\right)}^{\top },p{\left(\mathbf{x}\right)}^{\top }\}{\{a,b\}}^{\top },$$

(3)

where $R_i\left(\mathbf{x}\right)$ represents the RBFs associated with interpolation, and $p_j\left(\mathbf{x}\right)$ represents the polynomial terms used in conjunction with the RBFs to enhance stability and accuracy. In this formulation, N denotes the number of RBFs used, and m represents the number of polynomial terms in the spatial coordinates ${\mathbf{x}}^{\top }=[x,y]$.

The RBFs, $R_i\left(\mathbf{x}\right)$, are defined based on the Euclidean distance between the interpolation points, allowing the method to handle scattered data and complex geometries without requiring a structured grid. Some common choices for RBFs include Gaussian, Multiquadrics, Inverse Multiquadrics, and Thin Plate Splines, and the detailed descriptions of these RBFs can be found in [39,40]. The number of points n in the support domain of $\mathbf{x}$ directly influences the interpolation accuracy and determines the number of RBFs required for a specific problem.

The polynomial term, $p_j\left(\mathbf{x}\right)$, is often added to ensure the invertibility of the interpolation matrix, which is crucial for solving the resulting system of equations. Typically, $m\ll n$, the number of polynomial terms, is much smaller than the number of RBFs. However, including these monomial terms can enhance the accuracy of interpolation, particularly near the domain’s boundaries. When the polynomial terms are omitted (i.e., $m=0$), the method uses pure RBF interpolation, which can still provide accurate results for many applications, albeit with a higher sensitivity to the placement of nodes and the selection of the RBF’s shape parameter.

The coefficients $a_i$ and $b_j$ are unknowns that need to be determined by enforcing the interpolation condition at n distinct nodes within the support domain of the point of interest $\mathbf{x}$. These coefficients are typically computed by solving a system of linear equations formed by applying Equation (3) at the n nodes

$$u\left({\mathbf{x}}_k\right)=\sum _{i=1}^N{R}_i\left({\mathbf{x}}_k\right)a_i+\sum _{j=1}^m{p}_j\left({\mathbf{x}}_k\right)b_j,k=1,2,\dots ,n.$$

This results in a system of linear equations that can be expressed in matrix form

$$\mathbf{A}\mathbf{c}=\mathbf{f},$$

where $\mathbf{A}$ is the interpolation matrix, $\mathbf{c}={[a_1,a_2,\dots ,a_N,b_1,b_2,\dots ,b_m]}^{\top }$ is the vector of unknown coefficients, and $\mathbf{f}$ is the vector of known function values at the interpolation points. This matrix system can then be solved to determine the coefficients $a_i$ and $b_j$, providing the desired interpolated solution.

Figure 1 demonstrates the basic concept of RBFs in a scalar problem and how they are combined to form an interpolated function through RPI. The black dots on the x-axis represent the interpolation points $x_1,\dots ,x_6$, where the function $u\left(x\right)$ is evaluated. The combination of RBFs allows the method to flexibly adapt to scattered data points, making RPI an effective meshless technique for solving partial differential equations in domains with complex geometries. The green, blue, and red curves represent three RBFs centered at the interpolation points $x_2$, $x_4$, and $x_6$, respectively. The dashed vertical lines extend from each interpolation point to the corresponding value on the interpolated function, showing that the function passes through the exact values at these points, a key interpolation characteristic. These RBFs depend on the distance between x and their respective centers, providing a smooth approximation of the function. The thick black curve is the interpolated function $u\left(x\right)$, a weighted combination of the RBFs. In this case, the function is constructed as $u\left(x\right)=0.5·R_2\left(x\right)+0.3·R_2\left(x\right)+0.01·R_3\left(x\right)$, where the weights (coefficients) are determined by solving a system of linear equations based on the interpolation conditions at the points $x_1,\dots ,x_6$.

3. RPI Methodology for Fractional-Order Navier–Stokes Equations

Consider the time interval $[0,T]$ uniformly partitioned into M subintervals, where $\Delta t={\displaystyle \frac{T}{M}}$ and $t_k=k\Delta t$. Using the method in [41,42,43], Caputo fractional derivative operator (2) can be approximated by the quadrature formula

$$\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }u(x,t_k)& \approx D_{\Delta t}^{\alpha }u(x,t_k)\hfill \\ \hfill & =b_{\alpha ,\Delta t}\left(u(x,t_k)-\sum _{j=1}^{k-1}({\omega }_{\alpha ,k-j-1}-{\omega }_{\alpha ,k-j})u(x,t_j)-{\omega }_{\alpha ,k-1}u(x,t_0)\right),\hfill \end{array}$$

(4)

where  

$${\omega }_{\alpha ,j}:={(j+1)}^{1-\alpha }-j^{1-\alpha },j=1,2,\dots ,k,$$

and  

$$b_{\alpha ,\Delta t}:={\displaystyle \frac{\Delta t^{-\alpha }}{\Gamma (2-\alpha )}}.$$

Moreover, the truncation error is also determined by (see [41])

$$\left|{\mathcal{D}}_t^{\alpha }u(x,t_k)-D_{\Delta t}^{\alpha }u(x,t_k)\right|\le c_{\alpha }\Delta t^{2-\alpha }\underset{0\le t\le t_k}{max}\left|{\displaystyle \frac{{\partial }^{2}u(x,t_k)}{\partial t^2}}\right|,$$

where $c_{\alpha }$ is a constant that depends only on $\alpha$. This indicates that the scheme has an order of $\Delta t^{2-\alpha }$, provided the function u is twice continuously differentiable in t.

Using Equation (4), the discretized system of nonlinear algebraic equations corresponding to Equation (1) is

$$b_{\alpha ,\Delta t}A\sum _{k=1}^n{\omega }_{\alpha ,k}(U^{n-k+1}-U^{n-k})+B^n{U}^n=F^n,$$

(5)

where

$$A=\left(\begin{array}{ccc}I& 0& 0\\ 0& I& 0\\ 0& 0& I\end{array}\right),B=\left(\begin{array}{ccc}{B}_{11}& 0& B_{13}\\ 0& B_{22}& B_{23}\\ B_{31}& B_{32}& 0\end{array}\right),F=\left(\begin{array}{c}{f}_x\\ f_y\\ 0\end{array}\right),U=\left(\begin{array}{c}\widehat{u}\\ \widehat{v}\\ \widehat{p}\end{array}\right),$$

$$\begin{array}{}\mathrm{(6a)}& \hfill {\phi }_{ij}(u,v)& =u\left(x_i\right){\varphi }_{j,x}\left(x_i\right)+v\left(x_i\right){\varphi }_{j,y}\left(x_i\right)-{\displaystyle \frac{1}{Re}}({\varphi }_{jxx}\left(x_i\right)+{\varphi }_{jyy}\left(x_i\right)),\hfill \mathrm{(6b)}& \hfill B_{11}& =\left[{\phi }_{ij}(u,v)\right],B_{13}=\left[{\varphi }_{j,x}\left(x_i\right)\right],B_{22}=B_{11},\hfill \mathrm{(6c)}& \hfill B_{23}& =\left[{\varphi }_{j,y}\left(x_i\right)\right],B_{31}=B_{13},B_{32}=B_{23},\hfill \mathrm{(6d)}& \hfill F_x& ={[f_{x1},f_{x2},\dots ,f_{xN}]}^{\top },F_y={[f_{y1},f_{y2},\dots ,f_{yN}]}^{\top },\hfill \mathrm{(6e)}& \hfill \widehat{U}& =[\widehat{u_1},\widehat{u_2},\dots ,\widehat{u_N}],\hfill \mathrm{(6f)}& \hfill \widehat{V}& =[\widehat{v_1},\widehat{v_2},\dots ,\widehat{v_N}],\hfill \mathrm{(6g)}& \hfill \widehat{P}& =[\widehat{p_1},\widehat{p_2},\dots ,\widehat{p_N}],\hfill \end{array}$$

in which I is the $N\times N$ identity matrix and 0 is the $N\times N$ zero matrix.

Thus, the system can be rewritten as

$$b_{\alpha ,\Delta t}A(U^n-U^{n-1})+B^n{U}^n=-b_{\alpha ,\Delta t}\sum _{k=2}^n{\omega }_{\alpha ,k}(U^{n-k+1}-U^{n-k})+F^n,$$

(7)

such that at the first time level ($n=1$), it can be determined by

$$(b_{\alpha ,\Delta t}A+B^1)U^1=b_{\alpha ,\Delta t}A{U}_0+F^1,$$

(8)

and for $n\ge 2$, it yields

$$(b_{\alpha ,\Delta t}A+B^n)U^n=b_{\alpha ,\Delta t}A\left(U^{n-1}-\sum _{k=2}^n{\omega }_{\alpha ,k}(U^{n-k+1}-U^{n-k})\right)+F^n.$$

(9)

which can be expressed more conveniently as

$$U^n=G^n\left(U^{n-1}-\sum _{k=2}^n{\omega }_{\alpha ,k}(U^{n-k+1}-U^{n-k})\right)+{\left(b_{\alpha ,\Delta t}A\right)}^{-1}{F}^n,$$

(10)

with

$$G^n=b_{\alpha ,\Delta t}{(b_{\alpha ,\Delta t}A+B^n)}^{-1}A.$$

Algorithm 1 presents the proposed RPIM for solving (1). The algorithm starts by initializing parameters and setting up data points and centers. It then discretizes the time domain, establishes initial conditions, and computes the approximate solution. In each iteration, shape functions are updated, and the solution is refined for each time step. Finally, errors are calculated to assess the accuracy, and the algorithm outputs the approximate solution.

Algorithm 1 The proposed RPIM for solving two-dimensional conservation-form FNSE problem (1)

1: Initialize Parameters: Define all necessary parameters, including constants, coefficients, and initial settings for the problem.   2: Setup Data Points and Centers: Specify the number and types of data points and centers. Load these into the computational framework.   3: Define Time Discretization: Set the time discretization step $\Delta ={\displaystyle \frac{1}{\mathrm{num}\_\mathrm{times}}}$ and define the time interval $[0,1]$.   4: Set Initial Conditions: Establish the initial conditions for the problem.   5: Compute Exact Solution: Evaluate the exact solution at the data points for each time step.   6: Define Weight Function and Coefficients: Define the weight function $w_j={(j+1)}^{1-\alpha }-j^{1-\alpha }$ and compute $b_{\alpha ,\Delta }={\displaystyle \frac{{\Delta }^{-\alpha }}{\Gamma (2-\alpha )}}$.   7: Initial Approximation Solution: Compute shape functions $\Phi \left(x\right)$ and their derivatives at data points and centers. Formulate approximation matrix $M=b_{\alpha ,\Delta }A+B^1$. Calculate force function $F=b_{\alpha ,\Delta }A{U}_0+F^1$. Compute initial approximate solution $u_{\mathrm{approx}}={\Phi }^{\top }\left(M^{-1}F\right)$.   8: Iterate for Subsequent Time Steps: Repeat for $t=2,\dots ,\mathrm{num}\_\mathrm{times}$: Recompute shape functions $\Phi \left(x\right)$ and their derivatives. Update matrix $M=b_{\alpha ,\Delta t}A+B^n$. Update force function $F=b_{\alpha ,\Delta t}A(U^{n-1}-{\sum }_{k=2}^n{\omega }_{\alpha ,k}(U^{n-k+1}-U^{n-k}))+F^n$. Compute approximate solution $u_{\mathrm{approx}}={\Phi }^{\top }\left(M^{-1}F\right)$.   9: Compute Errors: Calculate the maximum error and RMS error on the evaluation grid: Maximum error: $maxerr=\parallel u_{\mathrm{approx}}-u_{\mathrm{exact}}{\parallel }_{\infty }$. 10: Output Results: Return the final approximate solution $u_{\mathrm{approx}}$.

4. Numerical Results

A series of numerical experiments assessed the proposed method’s accuracy and empirical convergence in solving two-dimensional Navier–Stokes equations. The computational implementation was performed using MATLAB R2019a on a system equipped with an Intel(R) Core(TM) i5-2410M 2.30 GHz processor and 4 GB memory. For this study, two test problems from the existing literature were selected to rigorously evaluate the approach’s performance. A thorough analysis of these examples demonstrates the effectiveness of the proposed method, with results presented in tabular and graphical formats. These numerical experiments showcase the method’s ability to handle various scenarios encountered in two-dimensional Navier–Stokes equations, providing insights into its practical applicability and computational efficiency.

Example 1.

Consider the following test problem for the FNSE (1):

$$\begin{array}{}\mathrm{(11a)}& \hfill u(x,y)& =& e^{-t}\left(2x^{2}y(1-x^2)(1-y)(1-2y)\right),\hfill \mathrm{(11b)}& \hfill v(x,y)& =& e^{-t}(-2x{y}^2(1-x)(1-2x){(1-y)}^2),\hfill \mathrm{(11c)}& \hfill p(x,y)& =& e^{-t}(x^2-y^2).\hfill \end{array}$$

Since this equation is challenging to solve analytically, a reverse engineering approach is proposed, where $u(x,y)$, $v(x,y)$, and $p(x,y)$ in Equation (11) are substituted directly into FNSE (1), and the source vector analytical is obtained as a function of $\alpha$ and time, i.e., $f(\alpha ;t)$, to evaluate the obtained numerical solutions. To conserve space, the obtained term, being highly nonlinear and lengthy, was omitted from this article. In addition, multi-quadric functions were used as shape functions in the RPI functions.

Figure 2 shows the Hammersley point set for interpolating scattered data. This quasi-random sequence of points achieves a uniform distribution across the domain while balancing computational efficiency and solution accuracy. The clear distinction between interior (blue circles) and boundary points (red crosses) demonstrates the method’s ability to deal with complex geometries and boundary conditions.

Figure 3 comprehensively visualizes the test problem solution. The top row contains 3D surface plots of the velocity components $(u,v)$ and pressure (p), which offer intuitive insights into the solution’s spatial distribution and overall behavior. The Absolute Error (AE) distribution for each variable is shown in the bottom row, allowing for an evaluation of the numerical solution’s accuracy compared to the exact solution. These error distributions highlight areas of high and low accuracy within the computational domain, guiding future numerical method refinements.

Table 1. The numerical results for the u, v, and p fields in Example 1 with $m=1$, $\alpha =0.75$, and $\Delta t=0.01$.

Node Density	Time Step	${\parallel \mathit{u}-{\mathit{u}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{v}-{\mathit{v}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{p}-{\mathit{p}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$
$4\times 4$	$\Delta t$	$3.4938\times {10}^{-4}$	$4.8785\times {10}^{-4}$	$2.2210\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{2}}$	$3.1950\times {10}^{-4}$	$4.0935\times {10}^{-4}$	$2.1570\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{4}}$	$3.0040\times {10}^{-4}$	$3.5558\times {10}^{-4}$	$2.0275\times {10}^{-4}$
$8\times 8$	$\Delta t$	$3.3457\times {10}^{-4}$	$4.1190\times {10}^{-4}$	$2.2721\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{2}}$	$3.2457\times {10}^{-4}$	$4.0380\times {10}^{-4}$	$2.0791\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{4}}$	$2.9563\times {10}^{-4}$	$9.4547\times {10}^{-5}$	$2.0287\times {10}^{-4}$
$16\times 16$	$\Delta t$	$2.2121\times {10}^{-4}$	$4.6207\times {10}^{-5}$	$2.8227\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{2}}$	$2.1790\times {10}^{-4}$	$4.4070\times {10}^{-5}$	$2.7673\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{4}}$	$2.0264\times {10}^{-4}$	$4.2373\times {10}^{-5}$	$2.6129\times {10}^{-4}$

presents the effect of mesh refinement and time step reduction on solution accuracy. For $m=1$ and $\alpha =0.75$, mesh refinement improves the accuracy in the u and v fields while slightly increasing the error in the p field. Reducing the time step from $\Delta t$ to $\Delta t/4$ generally enhances precision, though the improvement is not uniformly monotonic. Node density refers to the distribution and concentration of nodes within the computational domain, directly impacting the accuracy and convergence of the solution. It is used as a quantity aligned with the fixed-step size in finite difference methods, serving a similar role in controlling the discretization level of the problem. Mesh refinement consistently enhances accuracy in the u field and shows significant improvement in the v field. Conversely, the p field exhibits a slight increase in error as the node density decreases. Time step reduction generally improves accuracy, but the effect varies with field and node density. In finer meshes, decreasing the time step reduces the error across all variables, whereas in coarser meshes, some variables show substantial improvement with a smaller time step.

Table 2. The numerical results for the test problem in Example 1, using different numbers of polynomial basis functions, with $h=0.0625$, $\alpha =0.75$ and $\Delta t=0.01$.

Number of m	Node Density	${\parallel \mathit{u}-{\mathit{u}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{v}-{\mathit{v}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{p}-{\mathit{p}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$
	$4\times 4$	$6.0938\times {10}^{-4}$	$6.8785\times {10}^{-4}$	$2.2210\times {10}^{-4}$
$m=1$	$8\times 8$	$5.1950\times {10}^{-4}$	$5.0935\times {10}^{-4}$	$2.0570\times {10}^{-4}$
	$16\times 16$	$3.3040\times {10}^{-4}$	$3.5558\times {10}^{-4}$	$1.1275\times {10}^{-4}$
	$4\times 4$	$4.8792\times {10}^{-4}$	$5.4894\times {10}^{-4}$	$1.9232\times {10}^{-4}$
$m=2$	$8\times 8$	$2.7651\times {10}^{-4}$	$2.9812\times {10}^{-4}$	$1.5720\times {10}^{-4}$
	$16\times 16$	$1.4652\times {10}^{-4}$	$1.5618\times {10}^{-4}$	$1.0450\times {10}^{-4}$
	$4\times 4$	$3.7825\times {10}^{-4}$	$4.2658\times {10}^{-4}$	$1.5723\times {10}^{-4}$
$m=3$	$8\times 8$	$1.6352\times {10}^{-4}$	$1.7869\times {10}^{-4}$	$1.1124\times {10}^{-4}$
	$16\times 16$	$8.6543\times {10}^{-5}$	$9.1982\times {10}^{-5}$	$8.2310\times {10}^{-5}$

illustrates the influence of the number of polynomial basis functions on the results. As m increases from 1 to 3, a general trend of enhanced accuracy is observed, particularly for coarser meshes. The improvement is most notable when transitioning from $m=1$ to $m=2$, but the gains from $m=2$ to $m=3$ are less significant, especially for finer meshes. This suggests that higher-order polynomial approximations can significantly enhance solution quality, though there may be diminishing returns beyond a certain point.

Table 3. The numerical results for the test problem in Example 1, using different values of $\alpha$, with $m=1,h=0.25$ and $\Delta t=0.01$.

$\mathit{\alpha }$	Node Density	${\parallel \mathit{u}-{\mathit{u}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{v}-{\mathit{v}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{p}-{\mathit{p}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$
	$4\times 4$	$6.0938\times {10}^{-4}$	$6.8785\times {10}^{-4}$	$2.2210\times {10}^{-4}$
$\alpha ={\displaystyle \frac{1}{4}}$	$8\times 8$	$5.1950\times {10}^{-4}$	$5.0935\times {10}^{-4}$	$2.0570\times {10}^{-4}$
	$16\times 16$	$3.3040\times {10}^{-4}$	$3.5558\times {10}^{-4}$	$1.1275\times {10}^{-4}$
	$4\times 4$	$5.0012\times {10}^{-4}$	$5.6782\times {10}^{-4}$	$1.9854\times {10}^{-4}$
$\alpha ={\displaystyle \frac{1}{2}}$	$8\times 8$	$3.7825\times {10}^{-4}$	$4.2658\times {10}^{-4}$	$1.5723\times {10}^{-4}$
	$16\times 16$	$2.6543\times {10}^{-4}$	$2.8982\times {10}^{-4}$	$1.0450\times {10}^{-4}$
	$4\times 4$	$4.5009\times {10}^{-4}$	$5.2347\times {10}^{-4}$	$1.7523\times {10}^{-4}$
$\alpha ={\displaystyle \frac{2}{3}}$	$8\times 8$	$3.2218\times {10}^{-4}$	$3.7659\times {10}^{-4}$	$1.3872\times {10}^{-4}$
	$16\times 16$	$1.8785\times {10}^{-4}$	$2.1234\times {10}^{-4}$	$9.2123\times {10}^{-4}$
	$4\times 4$	$4.1093\times {10}^{-4}$	$4.7569\times {10}^{-4}$	$1.6345\times {10}^{-4}$
$\alpha ={\displaystyle \frac{3}{4}}$	$8\times 8$	$2.8652\times {10}^{-4}$	$3.2981\times {10}^{-4}$	$1.2984\times {10}^{-4}$
	$16\times 16$	$1.6432\times {10}^{-4}$	$1.8657\times {10}^{-4}$	$8.6543\times {10}^{-4}$

shows that while $\alpha$ influences error magnitude, it does not alter the error order regardless of the node density. The relationship between $\alpha$ and discretization parameters is complex, with certain $\alpha$ values improving accuracy for specific mesh resolutions and time steps.

In all scenarios, the error magnitudes for the pressure field (p) do not consistently exceed those for the velocity components (u and v). Mesh refinement improves the u and v fields more than the p field, which shows a slight increase in error for finer meshes at higher $\alpha$ values.

Example 2.

Consider the widely utilized benchmark problem known as decaying Taylor–Green vortices, which has an available analytical solution given by

$$\begin{array}{}\mathrm{(12a)}& & & u(x,y)=sin\left(x\right)cos\left(y\right)e^{{\displaystyle \frac{-2t}{Re}}},\hfill \mathrm{(12b)}& & & v(x,y)=-cos\left(x\right)sin\left(y\right)e^{{\displaystyle \frac{-2t}{Re}}},\hfill \mathrm{(12c)}& & & p(x,y)={\displaystyle \frac{1}{4}}(cos\left(2x\right)+cos\left(2y\right))e^{{\displaystyle \frac{-4t}{Re}}}.\hfill \end{array}$$

Similar to Example 1, a reverse engineering approach is proposed since solving these equations analytically is rigorous. In this approach, $u(x,y)$, $v(x,y)$, and $p(x,y)$ in Equation (12) are directly substituted into the FNSE (1), and the source vector is derived analytically as a function of $\alpha$ and time, i.e., $f(\alpha ;t)$, to evaluate the numerical solutions. It should noted that the highly nonlinear and lengthy term was omitted from this article to save space. Additionally, the RPI method employed multi-quadric functions as the shape functions.

Figure 4 provides a comprehensive visualization of the solution’s accuracy. The top row presents 3D surface plots of the velocity components ($u,v$) and pressure (p), revealing the spatial distribution and behavior of these fields. The velocity components exhibit sinusoidal patterns, with $u(x,y)$ showing alternating positive and negative regions corresponding to the vortex structure and $v(x,y)$ displaying a similar phase-shifted pattern with $\pi /2$ in both x and y directions. The pressure field $p(x,y)$ demonstrates the typical checkerboard pattern of Taylor–Green vortices, with high-pressure regions at vortex centers and low-pressure areas at vortex boundaries. The bottom row of Figure 4 shows each variable’s Absolute Error (AE) distribution, indicating the differences between numerical and exact solutions. The highest errors for the velocity components occur near domain boundaries and vortex centers, where gradients are steepest. The pressure error distribution is more complex, with higher errors corresponding to areas of rapid pressure change between vortices.

Table 4. The numerical results for u, v, and p fields in Example 2 when $m=1$, $\alpha =0.75$, $\Delta t=0.01$.

Node Density	Time Step	${\parallel \mathit{u}-{\mathit{u}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{v}-{\mathit{v}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{p}-{\mathit{p}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$
$4\times 4$	$\Delta t$	$6.0938\times {10}^{-4}$	$6.8785\times {10}^{-4}$	$2.2210\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{2}}$	$5.1950\times {10}^{-4}$	$5.0935\times {10}^{-4}$	$2.0570\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{4}}$	$4.3040\times {10}^{-4}$	$4.5558\times {10}^{-4}$	$1.5275\times {10}^{-4}$
$8\times 8$	$\Delta t$	$2.2457\times {10}^{-4}$	$2.1190\times {10}^{-4}$	$1.6791\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{2}}$	$2.1457\times {10}^{-4}$	$2.0190\times {10}^{-4}$	$1.4791\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{4}}$	$1.8040\times {10}^{-4}$	$1.7558\times {10}^{-4}$	$1.2275\times {10}^{-4}$
$16\times 16$	$\Delta t$	$2.8870\times {10}^{-4}$	$2.8869\times {10}^{-4}$	$3.1910\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{2}}$	$2.1830\times {10}^{-4}$	$2.1834\times {10}^{-4}$	$3.0191\times {10}^{-4}$
	${\displaystyle \frac{\Delta t}{4}}$	$2.0760\times {10}^{-4}$	$2.0510\times {10}^{-4}$	$3.0053\times {10}^{-4}$

demonstrates that refining spatial and temporal discretizations generally enhances solution accuracy, although the improvements are not uniform. As the spatial and temporal resolutions are increased, the accuracy of the numerical solution improves, providing a finer representation of the underlying physical phenomena. However, this enhancement in accuracy does not occur uniformly across all the node densities. For example, while the velocity components u and v improve significantly with finer discretizations, the pressure field p might show little improvement or even slight increases in error with finer meshes. This non-uniformity highlights the intricate nature of solving FNSE, where the interplay between spatial and temporal discretization must be carefully managed to optimize overall solution quality. Additionally, the computational cost associated with finer discretizations must be considered, as an increased resolution demands more processing power and memory. Therefore, achieving a balance between accuracy and computational efficiency is critical.

Table 5. The numerical results for u, v, and p fields in Example 2 using different numbers of polynomial basis functions (m) with $h=0.25$, $\alpha =0.75$ and $\Delta t=0.01$.

Number of m	Node Density	${\parallel \mathit{u}-{\mathit{u}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{v}-{\mathit{v}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{p}-{\mathit{p}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$
	$4\times 4$	$5.0938\times {10}^{-4}$	$5.8785\times {10}^{-4}$	$4.2210\times {10}^{-4}$
$m=1$	$8\times 8$	$4.1950\times {10}^{-4}$	$4.0935\times {10}^{-4}$	$4.0570\times {10}^{-4}$
	$16\times 16$	$2.8870\times {10}^{-4}$	$2.8869\times {10}^{-4}$	$3.1910\times {10}^{-4}$
	$4\times 4$	$4.1965\times {10}^{-4}$	$4.4835\times {10}^{-4}$	$4.1145\times {10}^{-4}$
$m=2$	$8\times 8$	$3.0417\times {10}^{-4}$	$3.3140\times {10}^{-4}$	$3.8991\times {10}^{-4}$
	$16\times 16$	$2.2457\times {10}^{-4}$	$2.1190\times {10}^{-4}$	$3.0791\times {10}^{-4}$
	$4\times 4$	$3.3040\times {10}^{-4}$	$3.7558\times {10}^{-4}$	$3.1910\times {10}^{-4}$
$m=3$	$8\times 8$	$3.0965\times {10}^{-4}$	$3.4835\times {10}^{-4}$	$3.1145\times {10}^{-4}$
	$16\times 16$	$2.2457\times {10}^{-4}$	$2.1190\times {10}^{-4}$	$3.0791\times {10}^{-4}$

illustrates the impact of polynomial basis functions on solution accuracy. Higher-order approximations yield slight improvements, regardless of the choice of node densities. As the order of the polynomial basis functions increases, the accuracy of the solution improves slightly. This effect is especially pronounced in coarser mesh settings, where higher-order polynomials capture finer details of the solution than lower-order approximations miss.

Table 6. The numerical results for u, v, and p fields in Example 2 using different fractional orders ($\alpha$) with $\Delta t=0.01$.

$\mathit{\alpha }$	Node Density	${\parallel \mathit{u}-{\mathit{u}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{v}-{\mathit{v}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$	${\parallel \mathit{p}-{\mathit{p}}_{\mathit{app}}\parallel }_{\mathbf{\infty }}$
	$4\times 4$	$8.9736\times {10}^{-4}$	$9.7654\times {10}^{-4}$	$5.3210\times {10}^{-4}$
$\alpha ={\displaystyle \frac{1}{4}}$	$8\times 8$	$5.2415\times {10}^{-4}$	$5.1144\times {10}^{-4}$	$4.1570\times {10}^{-4}$
	$16\times 16$	$2.4868\times {10}^{-4}$	$2.6389\times {10}^{-4}$	$3.0275\times {10}^{-4}$
	$4\times 4$	$7.1304\times {10}^{-4}$	$7.5735\times {10}^{-4}$	$4.2145\times {10}^{-4}$
$\alpha ={\displaystyle \frac{1}{2}}$	$8\times 8$	$3.3406\times {10}^{-4}$	$3.2145\times {10}^{-4}$	$3.5781\times {10}^{-4}$
	$16\times 16$	$1.3426\times {10}^{-4}$	$1.2351\times {10}^{-4}$	$3.1791\times {10}^{-4}$
	$4\times 4$	$5.4527\times {10}^{-4}$	$5.6652\times {10}^{-4}$	$3.8275\times {10}^{-4}$
$\alpha ={\displaystyle \frac{2}{3}}$	$8\times 8$	$2.2104\times {10}^{-4}$	$2.5668\times {10}^{-4}$	$3.4145\times {10}^{-4}$
	$16\times 16$	$9.3356\times {10}^{-5}$	$9.2415\times {10}^{-5}$	$3.1791\times {10}^{-4}$
	$4\times 4$	$4.4567\times {10}^{-4}$	$4.6789\times {10}^{-4}$	$3.6275\times {10}^{-4}$
$\alpha ={\displaystyle \frac{3}{4}}$	$8\times 8$	$3.1234\times {10}^{-4}$	$3.2901\times {10}^{-4}$	$3.3145\times {10}^{-4}$
	$16\times 16$	$2.8870\times {10}^{-4}$	$2.8740\times {10}^{-4}$	$3.1910\times {10}^{-4}$

highlights the impact of the fractional-order $\alpha$ on solution accuracy for time-fractional Navier–Stokes equations. Similar to the previous example, it shows that $\alpha$ affects error magnitude but does not change the error order regardless of node density. The interplay between $\alpha$ and discretization parameters is crucial, with certain values of $\alpha$ enhancing accuracy for specific mesh resolutions and time steps. This suggests the need for an adaptive approach to parameter selection to balance accuracy and computational efficiency.

Although the results from the RPI method were promising, some theoretical challenges emerged, particularly regarding the optimal choice of spatial form parameters, especially with a higher mesh resolution. Moreover, it is known that PRI methods perform poorly for highly irregular geometries, steep gradients, and discontinuities. They suffer from the curse of dimensionality in high-dimensional problems. The choice of appropriate support radius is a nontrivial task in compactly supported RBFs, where locality and overlap balance each other. The technique also requires heavy implementation in numerical integration and parameter tuning. Moreover, there is no theoretical guarantee of convergence. They are sensitive to multiple parameters requiring laborious tuning, which could be the major time-consuming factor affecting efficiency, accuracy, and robustness in practical engineering applications. For instance, the issues with pressure field calculations were present in all cases, indicating that further targeted research is required. The results emphasized that careful parameter selection is critical, as subtle interactions between the discretization parameters significantly impacted the solution quality. The findings highlight the importance of selecting computational parameters carefully to achieve the best accuracy, as the interaction between spatial and temporal discretization, polynomial order, and fractional order greatly affects the solution quality. Furthermore, further disadvantages of the proposed method have a very high computational cost and scalability problems for dense and ill-conditioned system matrices, which are difficult to reach with the increased number of nodes. Numerical instability and sensitivity of a solution may occur regarding shape parameters. Thus, their appropriate selection becomes critical but difficult in practice. Moreover, boundary conditions are imposed in an unnecessarily painful manner, especially while using globally supported RBFs.

5. Conclusions

This paper thoroughly evaluated the Radial Point Interpolation (RPI) method for time-fractional Navier–Stokes equations, demonstrating its cost-effectiveness and mesh-free approach for two-dimensional problems. Numerical trials, particularly focusing on the Taylor–Green vortex benchmark, highlighted the method’s performance and accuracy, identifying strengths and areas for improvement. The analysis revealed the intricate relationship between spatial and temporal discretization, polynomial order, and fractional order in determining the accuracy of a solution. While the RPI method showed promising results, it faces several challenges, particularly in selecting optimal spatial form parameters, handling irregular geometries, and managing high-dimensional problems. The method suffers from scalability issues, computational inefficiency, and numerical instability, with no theoretical convergence guarantee. Proper parameter tuning is crucial yet time-consuming, significantly impacting accuracy and robustness. These factors interact in complex ways, significantly influencing the overall quality of the numerical results. This highlighted the need for more specialized techniques or parameter selection adjustments to improve the method’s accuracy and stability.

Our future research will focus on refining the RPI algorithm to address the current limitations, particularly in pressure field calculations, aiming to further optimize its performance. This will likely involve developing more advanced error reduction techniques and enhancing the algorithm’s ability to handle complex, real-world fluid dynamics problems. The goal is to extend the RPI method’s applicability to a broader range of challenging scenarios, ultimately making it a more robust tool in computational fluid dynamics.