In this thesis, we focus on obtaining the numerical solution of singularly perturbed delay parabolic convection-diffusion problems in one and two-dimensions by some higher-order parameter-uniform numerical methods. First, we consider a one-dimensional singularly perturbed delay convection-diffusion problem, and we use the implicit-Euler scheme to discretize the time derivative and the hybrid scheme to discretize the spatial deriva- tives on the Shishkin mesh. We study the convergence of the scheme in the maximum norm, which shows the method converges uniformly with almost second-order accuracy in space and first-order accuracy in time. To validate the theoretical findings, numerical experiments are carried out.
It is well-known that the classical central difference scheme applied on the uniform mesh fails to provide satisfactory numerical solution for singularly perturbed convection- diffusion problems until we use unacceptably large number of mesh-points in comparison with the perturbation parameterε. Therefore, special attention has to be paid to obtain second-order ε-uniformly convergent numerical solutions for singularly perturbed PDE having convection term. Due to this, we use the Richardson extrapolation technique, which gives second-order convergence not only in space, but also in time. Numerical experiments justify the theoretical results.
Then, we proceed towards two-dimensional parabolic PDEs. To solve the parabolic PDE, we use a fractional-step method along with the Richardson extrapolation tech- nique. Fractional-step method allows us to convert the 2D problem into two 1D prob- lems. As a consequence, one can obtain the numerical solution by solving only two tridiagonal matrices instead of a banded pentadiagonal matrix. By using the Richard- son extrapolation technique, we are able to obtain second-order accuracy in space and time. Numerical experiments are carried out to show the convergence rate.
Next, we consider the singularly perturbed 2D delay parabolic convection-diffusion problem. We use the classical upwind scheme to discretize the problem on the Shishkin mesh. Also, we derive the error estimate, which shows that the method applied on the model problem is first-order convergence in space and time. Next, we use the Bakhvalov- Shishkin mesh instead of the Shishkin mesh, and we observe that error bound obtained for the scheme applied on the Bakhvalov-Shishkin mesh is optimal. Then, we consider the same problem and we use the fractional-step method to convert the 2D delay problem into two 1D delay problems. We solve the resulting 1D delay problems by the classical upwind scheme. Though, it gives almost first-order accuracy in space and time, but it is very efficient from the computational point of view.
The rest of this thesis consists of seven chapters and is organized as follows:
InChapter 2, the singularly perturbed DPDEs of the form (1.4.1) is solved numeri- cally on the uniform mesh in the temporal direction and the piecewise-uniform Shishkin mesh in the spatial direction. To discretize the temporal and spatial derivatives, we used the implicit-Euler scheme and the hybrid scheme, respectively. The method converges with almost second-order accurate in space and first-order accurate in time. Numerical experiments are carried out to validate the theoretical error estimates. We also carried out numerical experiment for singularly perturbed semilinear delay parabolic problem of the form (1.4.2).
The Richardson extrapolation technique is discussed in Chapter 3, which improves the accuracy of the upwind finite difference scheme on the piecewise-uniform Shishkin mesh, applied to the singularly perturbed DPDEs of the form (1.4.1). We proved the- oretically that extrapolation provides second-order ε-uniform convergence in space as well as in time. The numerical results reveal the theoretical finding.
In Chapter 4, we solved a singularly perturbed 2D parabolic convection-diffusion problem of the form (1.4.3) on the uniform mesh in the temporal domain and a special piecewise-uniform Shishkin mesh in the spatial domains. First, we use a fractional- step method for the discretization of the time derivative, which gives a set of two 1D problems. Then, we use the upwind finite difference scheme to solve those 1D prob- lems. To obtain a better approximate solution, we used the Richardson extrapolation technique. Theoretically we have shown that the extrapolation method provides second- order ε-uniform convergence in space and time. Numerical experiments are carried out to validate theoretical findings.
Chapter 5 deals with a singularly perturbed two-dimensional DPDE of the form (1.4.4). We discretized the temporal domain with the uniform mesh and the spatial domains with a special piecewise-uniform Shishkin mesh. To discretize the continuous problem, we used the implicit-Euler scheme and the classical upwind scheme for the temporal and spatial derivatives, respectively. The method converges uniformly with first-order (up to a logarithmic factor) in space and first-order in time. Numerical experiments are carried to validate the theoretical findings.
Chapter 6 contains the characterization of the Bakhvalov-Shishkin mesh, for dis- cretization of the spatial domains while solving a singularly perturbed 2D delay parabolic convection-diffusion problem of the form (1.4.4). We formed the Bakhvalov-Shishkin mesh with the help of an appropriate mesh-generating function. To discretize the con- tinuous problem, we applied the implicit-Euler scheme and the classical upwind scheme for the temporal and spatial derivatives, respectively. We proved theoretically and nu- merically that the proposed scheme applied on the Bakhvalov-Shishkin mesh provides first-order accuracy in space as well as in time.
Chapter 7 is devoted to solve the singularly perturbed 2D delay parabolic convection-diffusion problems of the form (1.4.4) with the fractional-step method on the uniform mesh in the temporal direction and a special piecewise-uniform Shishkin mesh in the spatial directions. With the help of the fractional-step method, we ob- tained two 1D stationary problems, which we discretized further by the classical upwind scheme. It is theoretically proved that the proposed scheme converges first-order (up to a logarithmic factor) in space and first-order in time. Along with the analysis, we provided numerical examples, which verify the theoretical findings.
Finally, Chapter 8 contains the summary of the results highlighting the contribu- tions made in this thesis and also provides possible future scopes of the present works.
Extensive numerical examples are given in support of the theoretical results and to show the accuracy of the numerical scheme. Those examples are presented at the end of each chapter of the thesis.
CHAPTER 2
Uniformly Convergent Hybrid Numerical Scheme for Singularly Perturbed 1D Delay Parabolic
Convection-Diffusion Problems on Shishkin Mesh
This chapter studies the numerical solution of singularly perturbed one-dimensional delay parabolic convection-diffusion problems. Since the solutions of these problems exhibit regular boundary layers in the spatial variable, we use the piecewise-uniform Shishkin mesh for the discretization of the domain in the spatial direction and uniform mesh in the temporal direction. The time derivative is discretized by the implicit-Euler scheme and the spatial derivatives are discretized by the hybrid scheme. For the proposed scheme, the stability analysis is carried out and parameter-uniform error estimates are derived. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.
2.1 Introduction
Let Ωx = (0,1), Λt= (0, T], D = Ωx×Λt and Γ = Γl∪Γb∪Γr, where Γl and Γr are the left and the right sides of the rectangular domain Dcorresponding to x= 0 andx= 1, respectively, and Γb = Ωx ×[−τ,0], τ > 0. In this chapter, we consider the following class of singularly perturbed DPDEs with Dirichlet boundary conditions:
∂
∂t +Lε
u(x, t) =−c(x, t)u(x, t−τ) +f(x, t), (x, t)∈D, u(x, t) = φb(x, t), (x, t) ∈Γb,
u(0, t) = φl(t), on Γl ={(0, t) : 0 ≤t≤T}, u(1, t) = φr(t), on Γr ={(1, t) : 0 ≤t≤T},
(2.1.1)
where
Lεu(x, t) = −εuxx(x, t) +a(x)ux(x, t) +b(x, t)u(x, t),
0 < ε 1 is the singular perturbation parameter and τ > 0 is the delay parame- ter. The coefficients a(x), b(x, t), c(x, t), f(x, t) on D and the boundary, initial val- ues φl(t), φr(t), φb(x, t) on Γ, are sufficiently smooth and bounded functions, such that a(x)≥α >0, b(x, t)≥0 and c(x, t) is nonzero on D.
Under these assumptions, the solution of the IBVP (2.1.1) exhibits a regular bound- ary layer of width O(ε) along x = 1. The terminal time T is assumed to satisfy the condition T =kτ for some positive integer k.
The classical finite difference methods applied on the uniform mesh fail to provide satisfactory numerical solution for SPPs, until we use unacceptably large number of mesh-points in comparison with the perturbation parameterε. This drawback motivates to developε-uniform numerical methods for SPPs,i.e.,the order of convergence and the error constant are independent ofε.
The outline of this chapter is as follows: In Section 2.2, we obtain the bounds of the solution of the continuous problem. Section 2.3 deals with the piecewise-uniform Shishkin mesh and the hybrid numerical scheme. The main proof of convergence has been derived in Section 2.4. Section 2.5 deals with the singularly perturbed semilinear DPDE. Numerical results are presented in Section 2.6 and the chapter ends with Section 2.7 that summarizes the main conclusions.