In this section, we shall present the numerical results obtained by the proposed numer- ical scheme (2.3.2) for three test problems on the piecewise-uniform rectangular mesh DN,M = ΩNx ×ΛMt . In all the cases we perform the numerical experiments by choosing the constant ρ
0 = 4.2, ∆t = 0.8/N and p= 1/∆t.
Example 2.6.1. Consider the following singularly perturbed 1D linear delay parabolic IBVP:
ut−εuxx+ (1 +x(1−x))ux =u(x, t−1) +f(x, t), (x, t)∈Ωx×(0,2],
u(x, t) = φb(x, t), (x, t)∈Ωx×[−1,0],
u(0, t) = 0, u(1, t) = 0, t∈[0,2].
(2.6.1)
We choose the initial data φb(x, t) and the source function f(x, t) to fit with the exact solution
u(x, t) = exp (−t) (C1+C2x−exp (−(1−x)/ε)),
whereC1 = exp (−1/ε) and C2 = 1−exp (−1/ε).
As the exact solution of the problem (2.6.1) is known, we calculate the maximum pointwise error by
eN,∆tε = max
(xi,tn)∈DN,M
u(xi, tn)−UN,∆t(xi, tn) ,
for each ε, where u(xi, tn) and UN,∆t(xi, tn) denote the exact and numerical solutions obtained on the meshDN,M withN mesh-intervals in the spatial direction andM mesh- intervals in thet-direction, such that ∆t=T /M is the uniform mesh-size. We determine the corresponding order of convergence by
pN,∆tε = log2
eN,∆tε e2N,∆t/2ε
.
Now, for each N and ∆t, we define theε-uniform maximum pointwise error byeN,∆t = maxεeN,∆tε and the corresponding ε-uniform order of convergence by
pN,∆t= log2
eN,∆t e2N,∆t/2
.
The calculated maximum pointwise errors eN,∆tε and the corresponding order of conver- gence pN,∆tε for Example 2.6.1 by using the hybrid scheme are presented in Table 2.1, for various values of ε and N. From the result, given in Table 2.1, one can observe the ε-uniform convergence of the scheme. We can observe that the numerical order of convergence does not clearly reflect the actual theoretical order of convergence of the proposed scheme (2.3.2) in space as proved in Theorem 2.4.4. The reason behind this, is the error consists of two parts due to spatial and temporal discretizations. The order of convergence observed in Table 2.1 is due to the effect of time error, which is first-order.
In order to justify the spatial order of convergence precisely, we have done the numer- ical experiments by taking M =N2 and displayed the maximum pointwise errors and the corresponding order of convergence in Table 2.2 for both inner and outer regions.
Also one can observe from this table that the order of convergence in nearly two in the outer region and close to 1.5 in the inner region, this is because of the logarithmic factor appears in the error estimate in the inner region.
Example 2.6.2. Consider the following singularly perturbed 1D linear delay parabolic IBVP:
ut−εuxx+ (2−x2)ux+xu=u(x, t−1) + 10t2exp(−t)x(1−x), (x, t)∈Ωx×(0,2],
u(x, t) = 0, (x, t)∈Ωx×[−1,0],
u(0, t) = 0, u(1, t) = 0, t∈[0,2].
(2.6.2)
As the exact solutionu(x, t) of the DPDE (2.6.2) is unknown, to obtain the pointwise errors and to verify theε-uniform convergence of the proposed scheme, we use the double mesh principle which is described as follows:
Let Ue(xi, tn) be the numerical solution obtained on the fine mesh De2N,2M = Ωe2Nx ×Λ2Mt with 2N mesh-intervals in the spatial direction and 2M mesh-intervals in the temporal direction, whereΩe2Nx is a piecewise-uniform Shishkin mesh as like ΩNx with transition parameter ρegiven by
ρe= min 1
2, ρ
0εln N
2
,
such that for i= 0,1,· · · , N, the i-th point of the mesh ΩNx coincides with 2i-th point of the mesh Ωe2Nx . Then for each ε, we calculate the maximum pointwise error by
EεN,∆t= max
(xi,tn)∈DN,M
U(xi, tn)−U(xe i, tn) , and the corresponding order of convergence by
PεN,∆t = log2
EεN,∆t Eε2N,∆t/2
.
For each N and ∆t, the quantities EN,∆t and PN,∆t are defined like eN,∆t and pN,∆t based on the errorEεN,∆t as in the previous example.
In Table 2.4, we have displayed the computed maximum pointwise errors EεN,∆t and the corresponding order of convergence PεN,∆t for Example 2.6.2. The ε-uniform convergence of the scheme can also be seen from Table 2.4. Further, we have computed the maximum pointwise errors and the corresponding order of convergence in Table 2.5 by consideringM =N2, to justify the spatial order of convergence properly for Example 2.6.2.
Example 2.6.3. Consider the following singularly perturbed semilinear delay parabolic IBVP:
ut−εuxx + (1 +x(1−x))ux+ exp(u) =u(x, t−1) +f(x, t), (x, t)∈Ωx×(0,2],
u(x, t) =φb(x, t), (x, t)∈Ωx×[−1,0],
u(0, t) = 0, u(1, t) = 0, t∈[0,2].
(2.6.3) We choose the initial data φb(x, t) and the source function f(x, t) to fit with the exact solution
u(x, t) = exp (−t) (C1+C2x−exp (−(1−x)/ε)), whereC1 = exp (−1/ε) and C2 = 1−exp (−1/ε).
By using the Newton’s linearization process (2.5.2) on (2.6.3), we obtain the following singularly perturbed linear system of delay IBVPs:
um+1t −εum+1xx + (1 +x(1−x))um+1x + exp(um)um+1 =um+1(x, t−1)−exp(um)(1−um) +f(x, t), (x, t)∈Ωx×(0,2], um+1(x, t) = φb(x, t), (x, t)∈Ωx×[−1,0],
um+1(0, t) = 0, um+1(1, t) = 0, t∈[0,2].
(2.6.4) Now, for a fixedm, we solve (2.6.4) by using the numerical method discussed in Section 2.3. As a convergence criterion for the Newton’s linearization process, we use
max
(xi,tn)∈DN,M
Um+1(xi, tn)−Um(xi, tn)
≤δ, m≥0,
where Um(xi, tn) is the m-th iteration solution at the mesh-point (xi, tn) and δ is the tolerance value. Here, we used the tolerance value δ= 10−7.
As the exact solution of the problem (2.6.3) is known, for each ε, we calculate the maximum pointwise error by
ˆ
eN,∆tε = max
(xi,tn)∈DN,M
u(xi, tn)−UN,∆t(xi, tn) ,
where u(xi, tn) and UN,∆t(xi, tn) denote the exact and numerical solutions obtained on the mesh DN,M with N mesh-intervals in the spatial direction and M mesh-intervals in the t-direction, such that ∆t = T /M is the uniform mesh-size. We determine the corresponding order of convergence by
ˆ
pN,∆tε = log2
eˆN,∆tε ˆ e2N,∆t/2ε
.
Now, for each N and ∆t, we define theε-uniform maximum pointwise error by ˆeN,∆t = maxεeˆN,∆tε and the corresponding ε-uniform order of convergence by
ˆ
pN,∆t= log2
eˆN,∆t ˆ e2N,∆t/2
.
The calculated maximum pointwise errors ˆeN,∆tε and the corresponding order of con- vergence ˆpN,∆tε for Example 2.6.3 by using the hybrid scheme are presented in Table 2.7, for various values of ε and N. The ε-uniform convergence of the scheme can be seen from Table 2.7. To justify the spatial order of convergence properly for Example 2.6.3, we have computed the maximum pointwise errors and the corresponding order of convergence in Table 2.8 by considering M =N2.
In order to show that the proposed method performs better than the earlier methods available in the literature, we have solved the problems given in Examples 2.6.1, 2.6.2 and 2.6.3 by using the first-order upwind scheme given in [32]. The corresponding numerical results are given in Tables 2.3, 2.6 and 2.9. One can compare the errors and the order of convergence given in Tables 2.1 and 2.3 for Example 2.6.1 and Tables 2.4 and 2.6 for Example 2.6.2 and Tables 2.7 and 2.9 for Example 2.6.3 and easily conclude that the proposed scheme performs better than the scheme used in [32].
To show the effect of some Shishkin-type meshes, we have computed the maximum pointwise errors and the corresponding order of convergence for the Shishkin mesh, the Bakhvalov-Shishkin mesh and the modified Bakhvalov-Shishkin mesh in Table 2.10 by taking M =N2.
To visualize the appearance of the boundary layers in the solutions of Examples 2.6.1, 2.6.2 and 2.6.3, we have plotted the surface plots for N = 64 in Figure 2.2 for ε= 10−4. In order to reveal the numerical order of convergence, we have plotted the maxi- mum pointwise errors in loglog scale, obtained by applying both the hybrid and upwind schemes for Examples 2.6.1, 2.6.2 and 2.6.3 in Figure 2.3, which again ensure the effec- tiveness of the proposed hybrid scheme in comparison with the upwind scheme.
Further, we have plotted the maximum pointwise errors in Figures 2.4 for all the examples by taking M = N2. From these figures one can easily observe the second- order convergence of the hybrid scheme, for the spatial variable.