We begin the thesis with a general introduction along with the goal and the motivation for solving singularly perturbed delay parabolic PDEs. We use the classical wind-up scheme to discretize the spatial derivatives on the standard Shishkin mesh and the Bakhvalov-Shishkin mesh.
Brief Background
If there is no delay in the highest order derivative in a DDE, such a type of DDE is called a retarded type. If the delay appears in the highest-order derivative, this type of DDE is called neutral type.
Objective and Motivation
This technique was used for singularly perturbed convection-diffusion BVP on the piecewise uniform Shishkin mesh by Natividad and Stynes [73] and on uniformly distributed mesh by Das and Natesan [16], while Mohapatra and Natesan [62] applied this technique for singularly perturbed convective-diffusion BVPs. delay of BVPs for ODEs. Erdogan [25] proposed an implicit finite difference scheme on the special Bakhvalov network for an IVP.
Some Notations and Terminologies
Shishkin-type meshes
Let ϕ be the generating function of the lattice on [0, xN/2] with the properties ϕ(0) = 0 and ϕ(1/2) = lnN, where ϕ(σ) is a continuously increasing and piecewise monotonic function differentiable. Furthermore, we assume that φ0 does not decrease, which leads to a lattice that does not condense to [0, ρ] as we move away from the layer.
Model Problems
Singularly perturbed 1D delay parabolic convection-diffusion
Singularly perturbed 1D semilinear delay parabolic problem
Singularly perturbed 2D parabolic convection-diffusion problem . 13
General Outline of the Thesis
In Chapter 2, the singularly perturbed DPDE forms (1.4.1) are numerically solved on a uniform grid in the time direction and a piecewise-uniform Šiškin grid in the spatial direction. In Chapter 4, we solved a singularly perturbed 2D parabolic convection-diffusion problem of the form (1.4.3) on a uniform grid in the time domain and a special piecewise-uniform Shishkin grid in the spatial domains.
Bounds for the Solution of the Continuous Problem
Decomposition of the solution
The following theorem will provide bounds for the derivatives of the smooth and singular components of the solution (2.1.1). We will use the following step-by-step procedure to prove the bounds for the derivatives of the smooth component w(x, t).
The Numerical Solution
The piecewise-uniform Shishkin mesh
By following the similar procedure as in the above three cases, one can get the bound for the other derivatives, i.e. for 0 ≤l+m≤5. Similarly, the necessary bound can be obtained for t≥2τ and the proof is therefore complete.
The finite difference scheme
Similarly, we define the left and right boundary points by ΓNl = DN,M ∩Γl and ΓNr = DN,M ∩Γr, respectively.
Convergence Analysis
We will now give our main result for the ε-uniform convergence of the numerical solution. Now, using the estimates of the derivatives of the smooth component given in Theorem 2.2.4 together with the inequality ≤2N−1, hi+hi+1 ≤4N−1, we obtain that
Semilinear Delay Parabolic Problem
Numerical results of the delay IBVPs of the form (2.1.1) and (2.5.2) are given in the following section.
Numerical Results
In Table 2.4, we have shown the maximum errors calculated at the point EεN,∆t and the corresponding order of convergence PεN,∆t for example 2.6.2. To justify the spatial order of convergence properly for Example 2.6.3, we have calculated the maximum point errors and the corresponding order of convergence in Table 2.8 considering M =N2.
Conclusion
To show the effect of several Shishkin-type meshes, we have calculated the maximum point 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. We show that the Richardson extrapolation technique improves the convergence order by O(N−2ln2N + ∆t2).
Introduction
Then, the Richardson extrapolation technique is applied to the computed solution by adjusting the transition parameter of the Shishkin lattice in order to improve the accuracy. Numerical experiments are performed to show the accuracy and efficiency of the proposed method.
The Numerical Solution
Discretization of the domain
Here we first discretize the domain with the uniform mesh in the time direction and by the piecewise uniform Shishkin mesh in the spatial direction. Here we construct the piecewise uniform mesh in such a way that the density of the mesh points is more in the boundary layer region than the outer region.
Numerical scheme
Error estimate for the difference scheme (3.2.2)
Richardson Extrapolation Technique
Error estimate for the extrapolated solution (3.3.3)
The section gives the estimation of the ε-uniform error of the numerical solution after the extrapolation technique. The main result of this chapter on the ε-uniform convergence of the numerical solution is given at the end of this section.
Error estimate for the smooth component Y
Error estimate for the singular component Z
Let u be the solution of the continuous problem (3.1.1) and Uextp be the solution obtained via the Richardson extrapolation technique (3.3.3) by solving the discrete problem (3.2.2) on two masksDN,M2 and D2N,2M2. Let u be the solution to the continuous problem (3.1.1) and Uextp be the solution obtained via the Richardson extrapolation technique (3.3.3) by solving the discrete problem (3.2.2) on two meshes DN,M and D2N,2M .
Numerical Results
The calculated maximum pointwise errors and the corresponding order of convergence for Example 3.4.1 are presented in Table 3.1 for different values of ε and N. In Table 3.3 we have shown the maximum pointwise errors (before and after extrapolation) and the corresponding order of convergence for example 3.4.2.
Conclusion
The calculated maximum pointwise errors and the corresponding order of convergence for Example 3.4.3 are presented in Table 3.4 for different values of ε and N. Moreover, it can be observed that the maximum pointwise errors and the corresponding order of convergence have been improved after the extrapolation technique.
Time Semidiscretization
The semidiscrete problem
The following lemma provides the bounds for the derivatives of the smooth and singular components. The scheme consists of the implicit Euler scheme for the time derivative and the hybrid numerical scheme for the spatial derivatives.
Extrapolation of b u
Error estimate for b u extp t
The Discrete Problem
Discretization of the domain
Note that, if ρl= 1/2, l =x, y, then the grid is uniform and in this case the error estimates can be obtained in the classical way.
Numerical scheme
As a result, the difference operator given in (4.4.1) satisfies the following discrete maximum principle. Discrete maximum principle) Assume that the discrete functionΨi satisfies Ψi ≥ 0 at i = 0, N.
Error estimate for the fully discrete solution
From the above theorem it can be observed that the time order of convergence is one and the spatial order of convergence is almost one (up to a logarithmic factor). To improve the order of convergence, we apply the Richardson extrapolation technique to the discrete solution Ui,jn of the fully discrete scheme.
Extrapolation of U b
Extrapolation of the smooth component
Extrapolation of the singular component
Again, using the similar approach used in [89] to the result obtained in Lemma 4.5.5, we obtain. Letube is the solution of the continuous problem (4.1.1) and Uextp is the solution obtained with the Richardson extrapolation technique, solving the fully discrete scheme at the time level tn=n∆t in two grids GN,M and G2N,2M. Then, for 0< q <1, we have the following error associated with Uextp:. u−ubextpt +ubextpt −Ubextp+Ubextp−Uextp. 4.5.15) Then, using the stability of the fully discrete scheme, one can obtain that.
Numerical Results
The calculated maximum point errors and the corresponding order of convergence for Examples 4.6.1 and 4.6.2 are presented in Tables 4.1 and 4.2, respectively, for different values of ε and N. Furthermore, the maximum point-wise errors (before and after extrapolation) for Examples 4.6.1 and 4.6.2 is plotted in loglog scale in Figures 4.2 and 4.4 respectively.
Conclusions
The numerical scheme used to discretize the federated problem consists of an implicit Euler scheme for the time derivative and a classical upwind scheme for the spatial derivatives. Then, to obtain a discrete problem, we use an implicit Euler scheme for the time derivative and a classical upwind scheme for the spatial derivatives.
Decomposition of the Solution and their Bounds
The outline of this chapter is as follows: In section 5.2, the bounds for the components of the solution (5.1.1) were discussed. The following theorem gives the bounds for the derivation of the components of the solution to the problem (5.1.1).
Domain Discretization
Numerical scheme
The finite difference operator (δt− +LNε ) defined in (5.3.1) satisfies the following discrete maximum principle on GN,M, which ensures ε-uniform stability of the difference operator (δt−+LNε ). Discrete Maximum Principle) Assume that the discrete function Ψni,j satisfies Ψni,j ≥0 on ΥN.
Error Analysis
Using the initial condition for the smooth part of the solution and the error estimate given in (5.4.1) together with the Taylor expansions, we obtain the truncation error for the difference equation (5.4.4), as The local truncation error for the difference equation (5.4.5) of the corner layer part E12 can be written as.
Numerical Results
The calculated maximum pointwise errors and the corresponding order of convergence for Examples 5.5.1 and 5.5.2 are presented in Tables 5.1 and 5.2, respectively, for various values of ε and N. To reveal the numerical order of convergence, we plotted the maximum pointwise errors (in loglog scale) in Figures 5.3 and 5.5, for Examples 5.5.1 and 5.5.2 respectively, again confirming the near first-order convergence of the scheme.
Conclusions
In Chapter 5 we saw that the upwind finite difference scheme on the Shishkin network for solving singularly perturbed 2D delay parabolic convection-diffusion IBVPs is almost first order (to a logarithmic factor) convergent, which is not optimal. We prove that the numerical scheme applied to the Bakhvalov-Shishkin network is first-order convergent in the discrete supremum norm, which is optimal and requires no additional computational effort compared to the standard Shishkin network.
Introduction
First, we discretize the spatial domain by the Bakhvalov-Shishkin network and the temporal domain by the uniform network. The outline of this chapter is as follows: Section 6.2 discusses the Bakhvalov-Sishkin mesh and the headwind scheme.
Domain Discretization
Numerical scheme
The finite difference operator (δt− +LNε ) defined in (6.2.2) satisfies the following discrete maximum principle on GN,M, which provides the ε-uniform stability of the difference operator (δt−+LNε ). Discrete maximum principle) Assume that the discrete function Ψni,j satisfies Ψni,j ≥0 on ΥN. Then (δt−+LNε )Ψni,j ≥0 on GN, M implies that Ψni,j ≥0at every point of GN, M.
Error Analysis
For the smooth component G, the truncation error can be written as (δ−t +LNε )(G (xi, yj, tn)−Gni,j). Following a similar approach made in the previous case for the boundary layer part E1, we can obtain
Numerical Results
The calculated maximum pointwise errors and the corresponding convergence order for examples 6.4.1 and 6.4.2 are shown for the Bakhvalov-Shishkin network in Tables 6.1 and 6.2, respectively, for different values of εandN. In fulfillment of this observation, we have plotted the maximum pointwise errors on the loglog scale for examples 6.4.1 and 6.4.2 in Figures 6.2 and 6.4.
Conclusions
We note that the order of convergence obtained on the Bakhvalov-Shishkin lattice is better than that obtained on the Shishkin lattice, which verifies the estimate proved in Theorem 6.3.1. Indeed, we can conclude that the upwind finite difference scheme on the Bakhvalov-Shishkin mesh outperforms the Shishkin mesh, regardless of the perturbation parameter ε.
Time Semidiscretization
The semidiscrete problem
Therefore, using the same argument as in the previous time interval, we can prove that|κ| ≤C. Following the same way as for the previous time interval (0, τ), we can obtain the required limit.
The Discrete Problem
Discretization of the domain
Numerical scheme
As a consequence, the difference operator given in (7.3.1) satisfies the following maximum discrete principle. Discrete Maximum Principle) Assume that the discrete function Ψi satisfies Ψi ≥ 0 at i = 0, N.
Convergence Analysis
Now, before proceeding further, we give some technical lemmas that will be required for the error bound. Then, applying the Taylor expansion formula and using the limit of the singular component, we get
Numerical Results
Tables 7.1 and 7.2 display the maximum pointwise errors and the corresponding orders of convergence of the calculated solution U for Examples 7.5.1 and 7.5.2, respectively. As a supplement to these observations, Figures 7.2 and 7.4 display the plot of N versus the maximum pointwise errors in loglog scale for both Examples 7.5.1 and 7.5.2, respectively.
Conclusions
It is proved theoretically and numerically that the order of convergence of the classical upwind scheme improves from almost first order to almost second order after extrapolation. Then, a classical upwind scheme is applied on partially uniform Shishkin grids in spatial directions for 1D stationary problems resulting from the first step.
Future Scope
A robust numerical scheme for singularly perturbed delay parabolic initial boundary value problems on equally spaced grids. Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid matching strategy.
Shishkin mesh in 1D
Visualization of the order of convergence through loglog plot
Visualization of the order of convergence through loglog plot
Visualization of the order of convergence through loglog plot for Example
Visualization of the order of convergence through loglog plot for Example
Shishkin mesh in the unit square
Visualization of the order of convergence through loglog plot for Example
Visualization of the order of convergence through loglog plot Example .2.119
Comparison of the order of convergence through loglog plot for Example
![Table 2.3: Maximum pointwise errors e N,∆t and the corresponding order of convergence p N,∆t for Example 2.6.1 by using the upwind scheme [32].](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10523576.0/64.892.126.786.393.920/table-maximum-pointwise-errors-corresponding-convergence-example-upwind.webp)
![Table 2.6: Maximum pointwise errors E N,∆t and the corresponding order of convergence P N,∆t for Example 2.6.2 by using the upwind scheme [32].](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10523576.0/66.892.126.786.393.919/table-maximum-pointwise-errors-corresponding-convergence-example-upwind.webp)
