A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

In this paper, we consider two dimensional (2D) Rayleigh-Stokes problem for a heated generalized second-grade fluid with fractional derivative and a nonhomogeneous term of the form: with initial and boundary conditions w(x, y, t) = g(x, y, t), (x, y) ∈ ∂ , w(x, y, 0) = h(x, y), (x, y) ∈ , (2) where 0 < γ < 1, = {(x, y)|0 ≤ x ≤ L, 0 ≤ y ≤ L}. The Rayleigh-Stokes problem has gained attention in recent years. This problem plays a vital role to show the dynamic behavior of some non-Newtonian fluids, and the fractional derivative in this model is used to capture the viscoelastic behavior of the flow [23,24].
Several numerical methods are presented in the literature for the solution of fractional Rayleigh-Stokes problem, for example, Chen et al. [25] have solved the problem using explicit and implicit finite difference methods, they have also presented its stability and convergence using Fourier analysis. The convergence order for both schemes is O(τ + x 2 + y 2 ). Ramy et al. [26] solved Rayleigh-Stokes problem using Jacobi spectral Galerkin method. The method they derived is efficient and easily generalizes to multiple dimensions. The advantages of this method are reasonable accuracy and relatively fewer degrees of freedom. Mohebbi et al. [27] used a higher-order implicit finite difference scheme for two-dimensional Rayleigh-Stokes problem and discussed its convergence and stability by Fourier analysis. The convergence order of their scheme is shown to be O(τ + x 4 + y 4 ).
High-order schemes produce more accurate results, but suffer from slow convergence due to the increase of complexity in the algorithm. Since explicit group methods reduce algorithm complexity [28][29][30][31], we propose the use of explicit group method for the solution of two-dimensional Rayleigh-Stokes problem for a heated generalized second-grade fluid. The main purpose of this article is to solve two-dimensional Rayleigh-Stokes problem with the high-order explicit group method (HEGM).
The paper is arranged as follows; in Sect. 2, we give the formulation of the high-order compact explicit group scheme, and its stability is discussed in Sect. 3. In Sect. 4, the convergence of the proposed scheme is discussed. In Sect. 5, some numerical examples are presented with discussion, and finally, the conclusion is presented in Sect. 6.

The group explicit scheme
First, let us define the following notations: where x = y = h = L M which represent the space step and t = T N represents the time step. The operators δ 2 x and δ 2 y , which consist of the three-point stencil [32], satisfy and The relationship between the Grunwald-Letnikov and Riemann-Liouville fractional derivatives is defined as [27,33] where ω 1-γ k are the coefficients of the generating function, that is, ω(z, γ ) = ∞ k=0 ω γ k z k . We consider ω(z, γ ) = (1z) γ for p = 1, so the coefficients are ω γ 0 = 1 and Let From (5) we can obtain the following: Using (3), (4), (7), (8), and (1), we have Multiplying both sides by τ (1 + 1 12 δ 2 x )(1 + 1 12 δ 2 y ), we have After simplifying and rearranging, we get Crank-Nicoslon (C-N) high-order compact scheme where Applying (8) to the group of four points (as shown in Fig. 1) will result in the following 4 × 4 system: where The matrix (9) is inverted to get the high-order compact explicit group equation where

Stability of the proposed method
First we recall the following lemma.

Lemma 1 ([34])
The coefficients η l satisfy the following relations: The stability of the proposed method is analyzed using the matrix analysis method. Form (9), we obtain

Proposition 1
The high-order explicit group scheme (12) is unconditionally stable.
Proof Let w k i,j and W k i,j be the approximate and exact solutions, respectively, for (1), and let k i,j = W k i,jw k i,j denote the error at time level k. Then from (11), where where I is the identity matrix and E is the matrix with unity values along each diagonal immediately below and above the main diagonal. Let ρ 1 , ρ 2 , and ρ 3 represent the maximum eigenvalues for M 1 , N 1 , and P 1 , respectively, then From (12), when k = 0, Supposing we will prove it for s = k + 1. Indeed, from (12) So, using matrix analysis via mathematical induction, we proved that the proposed method is unconditionally stable.

Theorem 1 The high-order explicit group scheme (10) is convergent with the order of convergence O(τ + h 4 ).
Proof From (18), we have Hence, we proved that the high-order explicit group scheme (10) is convergent with the order of convergence O(τ + h 4 ).

Numerical experiments and discussion
In this section, three numerical experiments were simulated using Core i7 Duo 3.40 GHz, 4 GB RAM and Windows 7 using Mathematica software. The acceleration technique "Successive over-relaxation (SOR)" is used with relaxation factor ω = 1.8 and convergence tolerance ζ = 10 -5 for the maximum error (L ∞ ); C 1 -and C 2 -order of convergence are used for space and time variables and calculated using [34] where h, τ and L ∞ represent the space-step, the time-step, and the infinity norm, respectively.

Example 2 ([27])
∂w(x, y, t) where 0 < x, y < 1, with initial and boundary conditions w(x, y, 0) = 0, and with the exact solution             Table 15 Computational complexity for the HEGM and C-N high-order finite difference method method Methods

Per iteration
Addition/subtraction Multiplication/division C-N (35 + 8(k -1))m 2 (13 + 4(k -1))m 2 HEGM (34 + 8(k -1))(m -1) 2 + (35 + 8(k -1))(2m -1) (10 + 4(k -1))(m -1) 2 + (13 + 4(k -1))(2m -1) The execution time, error, and number of iteration are shown for the comparison between standard point and HEGM from Table 1 to Table 6 Tables 1 to 6, respectively, and it can also be seen  Table 7 and Table 8 show the maximum errors and CPU timing at different values of γ 's for Example 1 and Example 2 respectively. Table 9 shows the maximum error at different values of the relaxation factor (ω's). Tables 10 to 14 represent the space and time variables' order of convergence for the HEGM, which show that the theoretical order of convergence is in agreement with the experimental order of convergence.    and 2, which show that the proposed method is effective and reliable. The comparison of execution timing between FEG (HEGM) and SP (C-N) for Example 1 and Example 2 are shown in Figure 6 and Figure 7 respectively, which depicted that HEGM method required less execution time as compared to the C-N. Figures 8 and 9 show the graphs of the maximum error using HEGM when γ = 0.5 and τ = 1 20 for Examples 1 and 2, respectively. The computational effort is shown in Tables 16 and 17; it can be seen that the HEGM re-quires fewer operations as compared to the high-order Crank-Nicolson finite difference method.

Conclusion
In this paper, we have solved two-dimensional fractional Rayleigh-Stokes problem for a heated generalized second-grade fluid using the HEGM. The C 2 -order of convergence shows that the theoretical order of convergence agrees with the experimental order of convergence. The proposed method reduces execution time and computational complexity as compared to the high-order compact Crank-Nicolson finite difference scheme. We proved the unconditional stability using the matrix analysis method; moreover, the proposed method is convergent.