Abstract
This paper presents adaptive, graded and uniform mesh schemes to approximate the solution of a fractional order advection-diffusion model, which generally shows a weak singularity at the initial time level. The temporal fractional derivative in the underlying problem is described in a Caputo form and is discretized by means of L1 scheme on a nonuniform mesh. The space derivative is discretized on a uniform mesh employing a fourth-order compact finite difference scheme. The adaptive grid is generated via equidistribution of a positive monitor function. Stability and convergence results for the proposed method on graded mesh are established. Numerical examples are provided to study the accuracy and efficiency of the proposed techniques and to support the theoretical results. A discussion about the advantages of the graded and adaptive meshes over the uniform one is also presented. The CPU times for the proposed numerical schemes are provided.
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1 Introduction
Fractional differential equation has emerged as strong tools in the study of various physical and biological phenomena and modelling of material system and financial processes, for example, see [1,2,3,4,5,6,7,8]. Such equations can be used to simulate practical phenomena more accurately then integer order one [32]. The advection-diffusion (AD) equation is employed in groundwater hydrology research to model the transport of passive tracers carried by fluid flow in porous medium [9] and in neurology [10]. It is also used to describe the transport dynamics in complex systems. In this study, we consider the following time-fractional advection-diffusion (TFAD) equation:
subject to the IC (initial condition)
and BCs (boundary conditions)
Here, a and b are real positive constants, \(f(x,t) \in C([0,1]\times [0,T])\) and \({g}(x) \in C[0,1].\) Further, \(D^{\alpha }_t\chi (x,t)\) denotes the Caputo derivative of order \(\alpha ,\) which is defined as [11]:
Equation (1) describes how the field variable \(\chi (x,t)\) in a medium varies under the influence of advection and diffusion processes. The solution of the problem considered has a weak singularity at \(t=0\). The regularities of the solution satisfy
where \(\hat{C}_1\) and \(\hat{C}_2\) are constants independent of t and x. In [50], the existence and uniqueness of the solution to the Caputo time-fractional diffusion equation with Dirichlet boundary condition have been investigated. The maximum principle was applied for proving the uniqueness result. Li and Wang [51] prove existence and uniqueness of the solution to the Caputo time-fractional convection diffusion reaction equation. Further, the reader can refer to [43, 52]. Due to the weak singularity and nonlocality character of the time-fractional operator, it is very difficult in obtaining the exact solution of time-fractional model problems. Many powerful computational techniques have been used in recent years by researchers to approximate the solutions of several time-fractional problems, for instance, see [33,34,35,36,37,38, 40, 41]. On the other hand, various numerical schemes were used for solving the TFAD equations. Zhuang et al. [13] designed an implicit meshless scheme for solving the time-dependent fractional AD equation with the Caputo time derivative. In this method, the L1 method is employed for approximation of Caputo temporal fractional derivative on uniform mesh, while an implicit meshless approach based on the moving least squares technique is employed for discretization of space derivative. Azin et al. [14] developed a hybrid numerical scheme based on Chebyshev cardinal functions and the modified Legendre functions to approximate the solution of (1) over a bounded time domain and an unbounded space domain. Li et al. [15] proposed a series of high-order numerical schemes on uniform mesh to solve Caputo-type advection-diffusion equation. The authors first constructed a series of high-order numerical algorithms to approximate the Caputo derivative and then derived a high-order finite difference scheme for solving Caputo-type advection-diffusion equation. In [16], Cao et al. presented a new high-order difference scheme on uniform mesh to solve Caputo-type AD equation. Mardani [17] proposed a meshless method, which is based on the moving least square (MLS) approximation, for solving a time-fractional advection-diffusion model with variable coefficients. In this approach, the time-fractional derivative (TFD) is approximated by a finite difference scheme on uniform mesh. It is important to point out that in [13,14,15,16,17], numerical schemes based on uniform mesh (simpler mesh) are designed to approximate the time-fractional derivative. Further, the weak singularity was not considered in these papers. The optimal rate of convergence in time direction was obtained by considering the exact solution which is smooth enough. Moreover, various numerical techniques were proposed for solving the time-fractional diffusion and reaction-diffusion problems, see [21,22,23,24,25,26,27,28,29,30,31] and their references. The authors of these papers ignored weak singularity at \(t=0\) and considered numerical examples with smooth analytical solutions to show that their methods have the optimal order convergence in the time direction. Furthermore, most of the above-stated methods are of lower orders of convergence in space direction.
In the current work, we aim to develop robust numerical techniques for solving (1)–(3) subjected to both smooth and nonsmooth analytical solutions. We derive graded and adaptive mesh numerical schemes for (1)–(3). In these methods, the Caputo time-fractional derivative is approximated by means of L1 scheme on nonuniform grids and the space derivatives are approximated by using a compact finite difference (CFD) scheme on uniform mesh. The graded and adaptive meshes on the time domain are constructed to overcome the weak singularity at \(t=0\), which produce a fine mesh near \(t=0\). The adaptive mesh is generated via equidistribution of a monitor function [18,19,20]. The theoretical results on the stability and convergence for the graded mesh technique are introduced. We consider three test problems to demonstrate the efficiency and accuracy of the suggested method and to support the theoretical results. The comparison between the results obtained with graded and adaptive meshes and those obtained with the uniform mesh is presented. The CPU times for the proposed techniques are provided. Numerical methods based on graded mesh or adaptive mesh were proposed in [42,43,44,45,46,47,48,49] to solve various kinds of boundary value problems for ordinary differential equation or partial differential equations.
The outline of this paper is as follows: Section 2 contains the description of the discretization scheme on the graded mesh. The adaptive mesh generation algorithm is described in Section 3. The proposed method on graded mesh is analyzed rigorously for the stability and convergence in Section 4. In Section 5, three test problems are solved and the numerical results are presented to show the robustness of proposed numerical algorithms. Finally, the conclusions are discussed in Section 6.
2 Derivation of a graded mesh numerical scheme
In this section, a graded mesh technique is derived for solving the TFAD model (1)–(3).
2.1 Time discretization
We discretize (1)–(3) over the domain [0, T], where \(T>0.\) Let \(t_m=T(m/\mathcal {N})^r,\)\(m=0,1,..., \mathcal {N}\) be the temporal grid points, where \(\mathcal {N}\) be a positive integer and r is the grading parameter. Let the temporal mesh size be \(\uptau _m=t_{m}-t_{m-1},\) \(m=1,2,..., \mathcal {N}\). If \(r=1,\) then the mesh is uniform. We approximate the Caputo TFD by employing the L1 scheme on the nonuniform mesh as follows
where \(\delta _t^-\chi (x,t_k)=\frac{\chi (x,t_k)-\chi (x,t_{k-1})}{\uptau _k}\) with \(\uptau _k=t_{k}-t_{k-1},\) \(\forall \) \(1\le k\le \mathcal {N} \) and \(\hat{\mathcal {R}}^m\) is the truncation error.
Lemma 1
([12]) Assume that the solution of TFAD problem satisfies (5). Then, we have the following bound for each \((x,t_m) \in \left( 0,1\right) \times \left( 0,T\right) \):
Considering (1) at \(t=t_{m}\) yields
Equations (2) and (3) can be expressed as follows
2.2 Spatial discretization
Here, we discretize (9)–(11) in space direction by means of a fourth-order CFD technique. We introduce uniform spatial grids with spatial step \(\triangle x\) on the interval [0, 1] such that \(\{0=x_{0}<x_{1}<..<x_n<.....<x_{\mathcal {M}}=1\}\), where \(x_n=n\triangle x,\) \(n=0,1,...,\mathcal {M}\) and \(\mathcal {M}\) is the number of mesh elements.
The second-order central finite difference approximation \({\delta }^2_{x}v(x_{n})\) for \(v''(x_{n})\) is defined by
The second-order central difference approximation \({\delta }_{x}v(x_{n})\) for \(v'(x_{n})\) is defined by
Denote \(F(x_n)=F_n,\) \(F'(x_n)=F'_n,\) \(v(x_{n})=v_{n},\) \(v'(x_{n})=v'_{n}\) and \(v''(x_{n})=v''_{n}.\)
Theorem 1
Suppose the solution v(x) belongs to the function space \(C^6[0,1].\) The fourth-order compact difference scheme for the problem
is given by
where \(p=-\frac{b^2}{12a}\) and \(q=-\frac{b}{12a}.\)
Proof
Inserting the Taylor’s series expansions for \(v_{n+1}\) and \(v_{n-1}\) into (12) and (13) yields
where
and
where
Using (16) and (17), we obtain the following difference approximation for (14) at \(x=x_{n}:\)
where
To obtain a fourth-order scheme, one needs to approximate \(v_{n}^{(3)}\) and \(v_{n}^{(4)}\) in (19). For this purpose, we differentiate (14) w.r.t. x and then set \(x=x_n\) to get
Further, differentiating twice (14) w.r.t. x and then setting \(x=x_n\) produces
By (22) and (20), it follows from (19) that
Using (16) and (17) in (23) gives
Inserting (24) into (18) produces the following fourth-order CFD approximation for the problem (14):
which completes the proof. \(\square \)
Now, let
At the point \((x_{n},t_{m})\), (26) leads to
where
When \(l=1,\) we obtain \(\beta _{m,1}={\uptau }_{m}^{-\alpha }.\) By (27), it follows from (9) that
By means of Theorem 1, equation (30) at the point \((x_{n},t_{m})\) can be written as
We denote \(\chi ^{m}_{n}=\chi (x_{n},t_{m})\) and \(f^{m}_{n}=f(x_{n},t_{m}),\ 0 \le m \le \mathcal {N}; 0 \le n \le \mathcal {M}\). Thus, by (28) and (31), one has
where \(\hat{\mathcal {R}}^m_n\) represents the truncation error at \((x_n,t_m),\) which is defined by
where \(\hat{C}\) is a positive constant. Equation (11) is discretized as
The IC (10) is discretized as
Denoting \(\hat{\chi }^{m}_{n}\) as an approximation of \(\chi ^{m}_{n}\) and neglecting \(\hat{\mathcal {R}}^{m}_{n}\) in (32) yields the following finite difference discretization for (1)-(3):
with
3 An adaptive numerical method
An adaptive mesh technique for solving the TFAD model (1)–(3) is presented in this section. We note that the graded mesh technique for solving the problem considered is defined by (32) and the complete discrete method based on adaptive grid can be obtained by altering the truncation error term in (32) with the truncation error term given in (42). The truncation error \(\bar{\mathcal {R}}^m\) for TFD in (7) relative to the adaptive mesh is defined by
Taking into account (40) and (31), we obtain
The truncation error \(\bar{\mathcal {R}}^m_n\) in (41) is defined by
where C denotes a positive constant. Denoting \(\bar{\chi }^{m}_{n}\) as an approximation of \(\chi ^{m}_{n}\) and neglecting \(\bar{\mathcal {R}}^{m}_{n}\) in (41) yields the following numerical scheme for (1)–(3):
with
3.1 Algorithm for adaptive mesh generation
Here we present an algorithm for generating the adaptive grid and for approximating the solution of (1)–(3) on the adaptive grid by employing (43)–(46).
Since the solution \(\chi (x,t)\) of the problem (1) shows a weak singularity at \(t=0\), a nonuniform adaptive time grid is generated by means of equidistribution of a positive monitor function, which is defined by (48). This kind of monitor function (48) has been employed in [18,19,20, 39]. Let \(\Theta ^{\mathcal {N}}=\{ 0=t_{0}<t_{1}<...<t_{m}<...<t_{\mathcal {N}}= T\}\) be the time mesh. The time mesh \(\Theta ^{\mathcal {N}}\) is called equidistributed if
The monitor function \(\hat{M}(\mu )\) in (47) is approximated by
In above equation, \(\delta _{t}^{2}\hat{\chi }_{n}^{m}\) denotes the central difference approximation of \(\hat{\chi }(x_n,t)\) on nonuniform temporal mesh. The following algorithm is proposed to solve (47):
Step I
Consider \(\hat{\jmath } =0,\) where \(\hat{\jmath } \) represents the iteration number. Take the uniform temporal mesh \(\Theta ^{\mathcal {M},\mathcal {N},(0)}= \{(x_{n},t_{m}^{(0)} ) \vert \ 0\le \ n\le \mathcal {M}, 0\le m\le \mathcal {N} \}\) as the initial value for the iteration. Go to the step II with \(\hat{\jmath } =0.\)
Step II
Solve (43)–(46) for \(\{ \bar{\chi }_{n}^{m,(\hat{\jmath } )} \}\) on \(\Theta ^{\mathcal {M},\mathcal {N},(\hat{\jmath } )}= \{ (x_{n},t_{m}^{(\hat{\jmath } )} ) \vert 0\le n\le \mathcal {M}, 0\le m\le \mathcal {N} \}\). Set \(\uptau _{m}^{(\hat{\jmath } )}=t_{m}^{(\hat{\jmath } )}-t_{m-1}^{(\hat{\jmath } )}\) for each m. Compute
and pick out \(\jmath \) such that
The monitor function \(\hat{M}_{n}^{k,(\hat{\jmath } )}\) in (49) was evaluated at the k-th grid point of the current grids. We set \( \hat{M}_{n}^{0,(\hat{\jmath } )}=\hat{M} _{n}^{1,(\hat{\jmath } )}\) and \(\hat{M}_{n}^{\mathcal {N},(\hat{\jmath } )}=\hat{M}_{n}^{\mathcal {N}-1,(\hat{\jmath } )}.\)
Step III
Choose a constant \(\hat{\psi }> 1\). If
then go to step V, else continue step IV.
Step IV
Set \(I_{m}^{(\hat{\jmath } )}=m\xi _{\jmath }^{\mathcal {N},(\hat{\jmath } )}/\mathcal {N}, \ m=0,1,..., \mathcal {N}.\) Interpolate \((I_{m}^{(\hat{\jmath } )},t_{m}^{(\hat{\jmath } +1)} )\) to \((\xi _{\jmath } ^{m,(\hat{\jmath } )},t_{m}^{(\hat{\jmath } )} )\). Generate a new mesh
Step V
Set \(\Theta ^{\mathcal {M},\mathcal {N},*}=\Theta ^{\mathcal {M},\mathcal {N},(\hat{\jmath } )}\) and \(\{ \bar{\chi }_{n}^{m,*} \} = \{ \bar{\chi }_{n}^{m,(\hat{\jmath } )} \},\) then stop.
Remark 1
It is observed that the coefficient matrix of (36)–(39) or (43)–(46) is strictly diagonally dominant with nonpositive off-diagonal elements and positive diagonal elements. Hence, the systems defined by (36)–(39) and (43)–(46) are solvable.
4 Stability and convergence
In this section, we study the stability and convergence for the numerical scheme (36)–(39).
4.1 Stability
Here, we present the stability bound of the present numerical scheme (36) for the considered time-fractional problem. We introduce \(L^{\infty }\)-norm for any mesh function \(U^m_n\), as follows
Lemma 2
The solution of (36) satisfies
for \(m=1,2,\dots ,\mathcal {N}\).
Proof
Fix \(m\in \{{1,2,...,\mathcal {N}}\}\). Choose \(n_0\) such that \(| \hat{\chi }^m_{i_{0}}|=\displaystyle \max _{0\le n\le \mathcal {M}}|\hat{\chi }^m_n|=||\hat{\chi }^m||_\infty \). Then, (36) at the mesh point \((x_{i_{0}},t_m)\) is
By taking \(L^{\infty }\)-norm in (55) one has
Equation (56) simplifies to
Thus, we get the desired result.\(\square \)
Lemma 3
The following properties hold for the coefficients \(\beta _{m,k}\) defined in (29):
Proof
Using the mean value theorem one can prove (i). Then, using (i) one can obtain (ii).
Let us define real numbers \(D_{m,i}\), for \(m=1,2,...,\mathcal {N}\) and \(i=1,2,...,m-1\) such that
In view of Lemma 3, it can be seen that \(D_{m,i}>0\) for all m and i.\(\square \)
Lemma 4
The solution of (36) satisfies
for \(m=1,2,\dots ,\mathcal {N}\).
Proof
We use mathematical induction on m to prove the lemma. For \(m=1,\) (54) reduces to
Thus, (58) is valid for \(m=1.\) Next, we assume that (58) holds true for all \(1\le m \le j-1,\) that is,
Now, we prove that the assertion (58) is valid for \(m=j.\) Considering (54) at \(m=j\), yields
Taking into account (59), it follows from (60) that
The above equation simplifies to
Now arranging the terms we get
In view of (57), the above equation can be written as
Equation (62) simplifies to
Thus, (58) is valid for \(m=j.\) Therefore, the assertion (58) is valid for all value of m. \(\square \)
Lemma 5
Let the parameter \(\lambda \) satisfy \(\lambda \le r\alpha \) and the real number \(D_{m,i}\) be defined by (57). Then, for \(1\le m\le \mathcal {N}\), we have
Proof
One can prove the lemma following the arguments used in Lemma 4.3 of [12].\(\square \)
Theorem 2
The solution of (36) satisfies
Proof
From lemma 4, we have
Setting \(\lambda =0\) in (63), one has
The above equation implies that
We now state and prove the main stability theorem. \(\square \)
Theorem 3
The numerical scheme defined by (36) is unconditionally stable.
Proof
Let \(\tilde{\chi }_n^m\) be the approximate solution of (36). The error \(\bar{e}_n^m=\tilde{\chi }_n^m-\hat{\chi }_n^m,\) \(n=0,1,..,\mathcal {M};\hspace{0.1cm}m=0,1,..,\mathcal {N}\) satisfies
Taking into account Lemma 2 and Theorem 2, one has
where \(||\bar{e}^m||_{\infty }=\displaystyle \max _{1\le n\le \mathcal {M}-1} |\bar{e}^{m}_{n}|.\) This demonstrates that the proposed numerical scheme (36) is unconditionally stable.\(\square \)
5 Convergence analysis
In this section, we study the convergence analysis of the numerical scheme based on graded mesh described by (36). Let \(e_n^m = \chi _n^m - \hat{\chi }_n^m\) for \(0 \le n \le \mathcal {M}\) and \(0 \le m \le \mathcal {N}\). Then, subtracting (36) from (33), one obtain the following error equation
where \(\hat{\mathcal {R}}^m_n\) is defined by (33). As the error terms at initial time level are zero, it follows from (69) that
Considering similar arguments as used in Lemma 2, we can obtain the following result
which is equivalent to
Lemma 6
The solution of (70) satisfies
Proof
We use induction on m to prove the result. When \(m=1,\) (72) reduces to
which suggests that (73) holds true for \(m=1\). Let’s assume that (73) holds true for \(1\le m \le j-1,\) that is
Now, we prove that (73) holds true for \(m=j.\) Considering (72) at \(m=j\) yields
Taking into account (74), it follows from (75) that
The last inequality is equivalent to
The above equation simplifies to
Thus, (73) is valid for \(m=j.\) Therefore, the conclusion of Lemma 6 is proved. We now state and prove the main convergence theorem.\(\square \)
Theorem 4
Let \(\chi (x,t)\) be the exact solution of (1)–(3) and \(\hat{\chi }_{m}^n\) be the discrete solution of (36)–(39). Then, there exist a constant \(C^{*}\) independent of \(\Delta x\) and \(\beta _{m,1}\) such that
Proof
From lemma 6, we have
Taking into account (63), it follows from (78) that
Hence, Theorem 4 is proved. \(\square \)
6 Numerical results
Here, three numerical examples of the form (1)–(3) are presented to illustrate the efficiency and robustness of proposed methods. It is worth mentioning that the exact solution to the first test problem has a weak singularity at the initial time \(t=0,\) while the solution of second one is smooth and the exact solution to the third problem is not known. We calculate the \(L_{\infty }\) norm error and the maximum \(L_{2}\) norm error in the computed solution corresponding to the graded mesh using the following formulae
and
where \({\hat{\chi }}_n^m\) and \(\chi (x_n,t_m)\) respectively denotes the computed solution and exact solution. We compare the numerical results obtained with the graded and adaptive meshes with the results obtained with the uniform mesh.
Example 1
Let us consider (1)–(3) with \(a=b=1,\) \({g}(x)=4x^2(1-x)^2\) and \(T=1\). The analytical solution is given by
The solution of above problem exhibits a weak singularity at \(t=0\). The right-hand side source function f(x, t) can be obtained by inserting (81) into left-hand side of (1).
The presented schemes are employed to approximate the solution of this problem for various values of \(\alpha ,\) \(\mathcal {N}\) and \(\mathcal {M}\). Figure 1 shows the formation of mesh points at final time level corresponding to the adaptive mesh technique and graded mesh technique for \(\alpha = 0.1, \,\, 0.4,\) and 0.6, when \(\mathcal {N}=\mathcal {M}=64.\) As it can be seen in Fig. 1 that the concentration of mesh points near \(t=0\) for \(\alpha =0.1\) is higher than that for \(\alpha =0.6.\) Figs. 2, 3 and 4 show the time evolution of mesh geometry on the adaptive mesh technique for \(\alpha = 0.1,\) \(\alpha = 0.4\) and \(\alpha =0.6,\) respectively. It can be noted from the figures that the number of iterations (NOI) increases as \(\alpha \) decreases. In particular, the NOI (within given tolerance) for \(\alpha =0.1,\) \(\alpha =0.4\) and \(\alpha =0.6\) are 24, 7 and 4 respectively. The 3D plots of the numerical results on graded, adapted and uniform grids for \(\alpha =0.1\) and 0.8 are depicted in Figs. 5 and 6, respectively. One can observe from the Figures that there is an initial layer in the solution profile which is consistent with (5). Further, one can observe from Figs. 5 and 6 that as \(\alpha \) decreases the layer at \(t=0\) becomes sharper.
Next, we calculate the \(L_{\infty }\) norm errors of presented schemes in time direction for different values of \(\alpha .\) Table 1 lists the \(L_{\infty }\) norm errors, the rate of convergence (ROC) and CPU time corresponding to the graded mesh with \(r=2(2-\alpha )/\alpha ,\) adaptive mesh and uniform mesh for \(\alpha =0.4,\,\, 0.6\) and 0.8. It can be noted from the tables that the scheme based on graded mesh yields much better accuracy (in temporal direction) as compared to the methods on adapted and uniform grids. Further, the method with adaptive mesh produces an approximation to the solution of the TFAD equation using more computational resources, both in terms of storage and CPU time. Moreover, we have calculated the errors on the graded mesh with \(r=(2-\alpha )/\alpha \) and \(r=(2-\alpha )/(2\alpha ),\) as listed in Tables 2 and 3, respectively for different values of \(\alpha \). One can observe from Tables 1, 2, and 3 that in the case of grading parameter \(r=(2-\alpha )/(2\alpha ),\) the rate of convergence is \((2-\alpha )/2,\) while for \(r=2(2-\alpha )/\alpha \) and \(r=(2-\alpha )/\alpha ,\) the optimal rate \((2-\alpha )\) is obtained. Further, the method on graded mesh with \(r=2(2-\alpha )/\alpha \) produces more accurate solution than the method with \(r=(2-\alpha )/\alpha .\) Furthermore, the uniform mesh method fails to provide an optimal \((2-\alpha )-\)th order of convergence in time.
Next, we calculate the convergence rates of proposed schemes in space with respect to \(L_{\infty }\) and \(L_2\) norm errors. To do so, we calculate the errors for various values of \(\mathcal {M}\) by fixing \(\mathcal {N}\) (viz. \(\mathcal {N}=12000\)). Table 4 lists the \(L_2\) norm and \(L_{\infty }\) norm errors and the rates of convergence obtained by the method on graded mesh with \(r=(2-\alpha )/\alpha \) for \(\alpha =0.8\). Table 5 lists the \(L_2\) norm and \(L_{\infty }\) norm errors and the rates of convergence obtained by the method on adapted mesh for \(\alpha =0.8.\) The tables indicate that the computed solution converges to the exact solution with fourth-order accuracy and confirm that the numerical results are in agreement with the theoretical results in Theorem 4. The \(L_{\infty }\) norm errors obtained on graded mesh with \(r=(2-\alpha )/\alpha ,\) adapted grid and uniform grid for \(\alpha =0.4,\,\, 0.6\) and 0.8, are depicted in Figs. 7, 8 and 9, respectively. From the figures, one can observe that the error decreases with the increase in \(\mathcal {M},\mathcal {N}\) and the scheme based on graded mesh yields much better accuracy as compared to the methods on adapted and uniform grids. The 3D plots of the absolute errors (in time) obtained by the methods on graded, adapted and uniform grids for \(\alpha =0.8\) are shown in Fig. 10 when \(\mathcal {M}=\mathcal {N}=64.\) It can be observed from the Figures that the error increases towards \(t=0\) and the present method with graded grid gives far better results as compared to the method with adapted grid or uniform grid.
Example 2
Consider (1)–(3) with \(a=b=1,\) \({g}(x)=4x^2(1-x)^2,\) and \(T=1\). The exact solution of this problem is \(\chi (x,t)=(2 x (1-x))^2 (t^{3+\alpha }+\sin (x)).\) This example has a smooth solution at \(t=0\).
The proposed scheme based on uniform mesh is employed to approximate the solution of this problem for several values of \(\alpha ,\) \(\mathcal {M}\) and \(\mathcal {N}\). The \(L_{\infty }\) errors for \(\alpha =0.4, 0.6\) and 0.8 are reported in Table 6. One can conclude from the table that the uniform mesh method has an optimal rate convergence (i.e., \((2-\alpha )\)) in time direction in the case when the exact solution to the TFAD problem is smooth.
Example 3
Consider (1)–(3) with \(a=b=1,\) \({g}(x)=\sin x,\) \(T=1\) and \(f(x,t)=(1+t^4)(x^2-\pi x)+t^2.\) The exact solution of this problem is not known.
The proposed schemes on graded mesh with \(r=(2-\alpha )/\alpha ,\) adapted mesh and uniform mesh are employed to approximate the solution of this problem for several values of \(\alpha ,\) \(\mathcal {M}\) and \(\mathcal {N}\). The \(L_{\infty }\) errors for \(\alpha =0.4, 0.6\) and 0.8 are reported in Table 7. One can observe from the table that the graded and adaptive mesh methods yield the optimal rate of convergence O(\(\mathcal {N}^{-(2-\alpha )}\)) in time, while the uniform mesh yields the suboptimal order convergence, that is, the order is close to \(\alpha \). Table 8 presents the \(L_{\infty }\) norm errors and the corresponding rates of convergence in space obtained by the methods on adapted mesh, graded mesh with \(r=(2-\alpha )/\alpha \) and uniform mesh for \(\alpha =0.8.\) The table indicates that the computed solution converges to the exact solution with fourth-order accuracy. The 3D plots of the numerical solutions on graded, adapted and uniform grids for \(\alpha =0.1\) are depicted in Fig. 11. One can observe from the Figure that there is an initial layer in the solution profile.
7 Conclusions
In this article, efficient and robust numerical schemes based on graded and adaptive meshes have been developed for solving the TFAD model with weakly singular solution. The temporal derivative is described in the sense of Caputo. We have constructed adaptive moving mesh algorithm and graded mesh technique to deal with the weak singularity at the initial time. The space derivative is discretized by a high-order difference scheme. It has been shown that the graded mesh method is unconditionally stable. Convergence result of the method based on graded mesh has been established. Three numerical examples were solved to demonstrate the applicability and efficiency of proposed methods. The computed results suggest that the method based on graded or adapted mesh well approximate the solution of a given TFAD problem and yields the optimal \((2-\alpha )-\)th order of convergence in time. The results obtained with the graded or adaptive mesh are better as compared to those obtained with the uniform mesh in terms of numerical accuracy. The uniform mesh method has the \(\alpha -\)th order of convergence in time in the case when the solution is nonsmooth. The method with adaptive grid produces an approximation to the solution of the TFAD problem using more computational resources. In the subsequent paper, we will design and analyze robust numerical scheme based on adaptive and graded meshes for the efficient numerical solution of a TFAD model with variable coefficients.
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The first author received financial support from NBHM, DAE under the project no. \( 02011/7/2023/NBHM (RP)/R \& D II/ 2877\).
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Pradip Roul: conceptualization, methodology, data curation, writing—original draft, software, investigation, validation. S. Sundar: methodology, validation.
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Roul, P., Sundar, S. Novel numerical methods based on graded, adaptive and uniform meshes for a time-fractional advection-diffusion equation subjected to weakly singular solution. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01804-0
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DOI: https://doi.org/10.1007/s11075-024-01804-0