# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
## 4 Numerical Results

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 \(\Delta t=0.01\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{\alpha}\) & **Node Density** & \(\|\mathbf{u}-\mathbf{u}_{app}\|_{\infty}\) & \(\|\mathbf{v}-\mathbf{v}_{app}\|_{\infty}\) & \(\|\mathbf{p}-\mathbf{p}_{app}\|_{\infty}\) \\ \hline \multirow{4}{*}{\(\alpha=\frac{1}{4}\)} & \(4\times 4\) & \(6.0938\times 10^{-4}\) & \(6.8785\times 10^{-4}\) & \(2.2210\times 10^{-4}\) \\  & \(8\times 8\) & \(5.1950\times 10^{-4}\) & \(5.0935\times 10^{-4}\) & \(2.0570\times 10^{-4}\) \\  & \(16\times 16\) & \(3.3040\times 10^\(v\) fields more than the \(p\) field, which shows a slight increase in error for finer meshes at higher \(\alpha\) values. Consider the widely utilized benchmark problem known as decaying Taylor-Green vortices, which has an available analytical solution given by equationparentequation  
\[u(x,y) =\sin(x)\cos(y)e^{\frac{-2t}{2\alpha t}}, \tag{12a}\] \[v(x,y) =-\cos(x)\sin(y)e^{\frac{-2t}{2\alpha t}},\] (12b) \[p(x,y) =\frac{1}{4}(\cos(2x)+\cos(2y))e^{\frac{-4t}{2\alpha t}}. \tag{12c}\]  
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 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 illustrates the impact of polynomial basis functions on solution accuracy.