Natesan Srinivasan, Associate Professor, Department of Mathematics, Indian Institute of Technology Guwahati for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for a degree. The scheme uses an appropriate combination of the upwind midpoint scheme and the classical central difference scheme for the spatial discretization and the backward Euler scheme for the time derivative discretization.
Brief Background
As a result, the perturbative expansion cannot satisfy all the boundary or initial conditions. Further discussion of norm selection can be found in Miller et al. [54].
Objective and Motivation
A few papers (see [12,14] ) dealing with nearly second-order uniformly convergent methods are available for singularly perturbed stationary convection-diffusion problems exhibiting boundary layers. Recently, in [2,60], Natesan and Bawa applied cubic spline method to obtain second-order uniformly convergent hybrid numerical schemes for singularly perturbed BVPs.
Preliminaries
Ifε≥N−1, then the model problems considered in the thesis are not difficult to solve computationally in practice. Let us consider the random meshes in the spatial and temporal directions respectively as 0 = x0 < x1 <.
Model Problems
Singularly perturbed parabolic convection-diffusion problems with bound-
Singularly perturbed parabolic convection-diffusion problems with in-
Singularly perturbed problems of mixed parabolic-elliptic type
Organization of the Thesis
This chapter proposes a hybrid numerical scheme for singularly turbulent one-dimensional parabolic convection-diffusion problems featuring a regular boundary layer. The numerical scheme consists of the classical backward-Euler method to approximate the time derivative and a hybrid finite difference scheme (a proper combination of the upwind mean scheme in the outer region and the classical central difference scheme in the boundary layer region ) for the discretization space.
Bounds on the Solution and its Derivatives
The Time Semidiscretization
- Discretization of the time domain
- The semidiscrete scheme
- Convergence analysis
- Asymptotic behavior of the solution of semidiscrete problem
To analyze the uniform convergence of the solution un(x) of (2.6) to the exact solution u(x, tn), we will perform the stability analysis and also derive the consistency result of the scheme (2.6).
The Spatial Discretization
The piecewise-uniform Shishkin mesh
The finite difference scheme
Here, the transition point 1−τ, which separates the coarse and fine parts of the mesh, is obtained by taking
Error analysis
Then they hold the following ratings:. x−s)ng(s)dje expression for the residue obtained from Taylor. To obtain appropriate estimates for the local truncation error τi,ˆun+1, proper bounds on the truncation error τi,xbu are derived in the following lemma.
Uniform Convergence of the Fully Discrete Scheme
Numerical Results
In table 2.2 we show the calculated maximum point-wise errors EεN,∆t and the corresponding order of convergence PεN,∆t for example 2.49. From the results given in Tables 2.1 and 2.2, we see the monotonically decreasing behavior of the calculated ε-uniform errors.
Conclusion
This chapter deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection-diffusion problems exhibiting a regular boundary layer. Consider the following 1D singular perturbed parabolic convection-diffusion IBVP, as discussed in the previous chapter, placed on the domain G= Ω×(0, T], Ω = (0,1).
Bounds on the Solution Decomposition
For all non-negative integers l, m satisfying 0 ≤ l+m ≤ 5, the smooth component v and the layer component w, defined in (3.5) and (3.6) respectively, satisfy the following bounds ∂l+mv. First, we need to obtain the stronger bounds on the smooth componentv, defined in (3.5) and its derivative.
Numerical Approximation
Discretization of the domain
The classical implicit upwind scheme
Extrapolation of U N,∆t
- Extrapolation technique
- Solution decomposition
- Extrapolation of V N,∆t
- Extrapolation of W N,∆t
- Convergence result of the solution U extp N,∆t
We now define the smooth component ηk as the restriction ηk∗ to the domain G, and the function ηk is therefore the solution to the following problem. Define the functions Fk, k = 1,2, as solutions to the following initial-boundary problems.
Numerical Results
Conclusion
In general, due to the presence of discontinuity in the convection coefficient (x), the solution u(x, t) of the problem may possess an inner layer of width O(ε) in the vicinity of the point x =ξ. But in fact the nature of the inner layer depends on the sign of the convection coefficient on either side of the line of discontinuity. The outline of this chapter is as follows: Section 4.2 provides a-priori bounds on the analytical solution and those on the derivatives of the solution via analysis.
Bounds on the Solution and its Decomposition
Here, the analysis is carried out separately to the subregions G−, G+ to obtain the stronger bounds on the smooth component v, defined in (4.7) and its derivatives. By henceforth applying the above arguments separately to the subregion G−, one can define g1(t) similarly and thus the required bounds on the smooth component v can be easily established. Now we proceed to find the required bounds on the layer component w, defined in (4.8) and its derivatives on the domain G−∪G+.
Numerical Approximation
Discretization of the domain
First, we define a piecewise uniform Shishkin lattice by dividing the spatial domain Ω into four subintervals as On the time domain [0, T], equidistant grids are introduced into the time variable such that.
The backward-Euler hybrid finite difference scheme
On each subinterval a uniform mesh with N/4 mesh intervals is placed such that. indicates the set of interior points of the mesh. where σ0 is a positive constant will be chosen later. On the time domain [0, T], the equidistant meshes are entered into the temporal variable such that. where M denotes the number of mesh intervals in the t-direction. After rearranging the terms in (4.14), we get the following form of the difference scheme on the meshGN,Mε.
Error Analysis
The main convergence result
The proof is divided into two different cases depending on the location of the mesh point xi ∈ΩN,εx. Here, the estimate of Uin−u(xi, tn) follows easily from Lemmas 4.4.3 and 4.4.6 by invoking the triangle inequality for the error. Moreover, if we use our approach to analyze the hybrid difference scheme for the stationary case, we can sharpen (4.52) to O(N−2ln2N).
Numerical Results
From the results given in Tables 4.1 and 4.2, we clearly see that the calculated ε-uniform errorsEN,∆t monotonically decrease as N increases. As a supplement to this observation, we have plotted the maximum point-wise errors for Examples 4.5.1 and 4.5.2 in Figure 4.3. Next, we see that the numerical results presented in Tables 4.1 and 4.2 do not clearly reflect the actual theoretical order of convergence of the proposed scheme (4.14) for the spatial variable, as predicted by Theorem 4.4.10.
Conclusion
Optimal error estimation of upwind scheme on Shishkin-type mesh for single-perturbed parabolic problems with discontinuous. Suitable conditions for the mesh generating functions are derived, which are sufficient for convergence of the method, uniformly with respect to the perturbation parameter. We consider once more the following class of uniquely perturbed parabolic IBVPs located on the domain G−∪G+:.
Numerical Approximation
Shishkin-type meshes
Then fix the transition points as in the Shishkin network, by choosing the transition parameters σ1 and σ2 as the following functions of N and ε. It is recognized that the properties of ψ allow easy characterization of the uniform convergence behavior of certain numerical methods for linear convection-diffusion problems with smooth data. Below we list some properties of the Shishkin-type meshes that will be used later in the analysis.
The implicit upwind finite difference scheme
It is easy to check whether the meshes S and B−Smesh satisfy the conditions given in (5.8) and (5.9). From [72], it can be shown that the finite difference operator LN,Mε satisfies the following well-known discrete maximum principle, which leads to ε-uniform stability of the difference operator LN,Mε.
Error Analysis
The main convergence result
Numerical Results
Since the exact solutions of the IBVPs (5.30) and (5.31) are not known, the numerical results of the winding scheme (5.13) are illustrated using the double mesh principle as described in Chapter 2. In Tables 5.2 and 5.3 for both the S mesh and the B−S mesh, we presented the calculated maximum pointwise errors EεN,∆t and the corresponding order of convergence PεN,∆t for Examples 5.4.1 and 5.4.2 respectively. Additionally, we have displayed the maximum pointwise errors over the full domain for Examples 5.4.1 and 5.4.2 in Figures 5.1 and 5.2, respectively.
Conclusion
This chapter deals with the study of a hybrid numerical scheme for a class of singly perturbed mixed parabolic-elliptic problems possessing both boundary and inner layers. In the first subdomain the given problem takes the form of parabolic reaction-diffusion type, while in the second subdomain elliptical convection-diffusion reaction problems arise. The proposed method is analyzed on a layer resolving piecewise uniform Shishkin mesh and is shown to be ε-uniformly convergent with near second-order spatial accuracy in the discrete supreme norm, provided that the perturbation parameter ε satisfies ε ≤N− 1.
Introduction
To solve these problems, the time derivative is discretized using the classical inverse Euler method. While for the spatial discretization of the problem, a classical central difference scheme is used on the first subdomain, and a hybrid finite difference scheme (a suitable combination of the upwind midpoint scheme in the outer regions and a classical central difference scheme in the inner layer regions) is proposed on the second subdomain.
Bounds on the Solution and its Decomposition
An immediate consequence of the above maximum principle is the following stability result, which implies the uniqueness of the solution of the IBVP. Before proving this theorem, the following result which will be used in Theorem 6.2.4 is stated below. Since the functionsvi−andvi+ are solutions to the problems (6.5) and (6.6) respectively, which are independent of the parameter ε, they therefore have the following ε-uniform bounded derivatives.
Numerical Approximation
Discretization of the domain
The backward-Euler hybrid finite difference scheme
After rearranging the terms in (6.16), we obtain the following form of the difference scheme on the grid GN,Mε.
Error Analysis
The main convergence result
Numerical Results
Conclusion
This chapter is devoted to developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection diffusion problems exhibiting a regular boundary layer. The numerical scheme consists of the Peaceman and Rachford alternating direction method for time discretization and a hybrid finite difference scheme (a proper combination of the midpoint upwind scheme in the outer region and the classical central difference scheme in the boundary layer region) for the spatial discretization. This is achieved by constructing a special rectangular mesh involving piecewise uniform Shishkin meshes in the spatial directions.
Bounds on the Solution Decomposition
The Time Semidiscretization
- Discretization of the time domain
- The semidiscrete scheme
- Convergence analysis
- Asymptotic behavior of the solutions of semidiscrete problems
This property is necessary to study the asymptotic behavior of the exact solutions of the semidiscrete problems as mentioned in [18]. This ensures that each step in the scheme has a unique solution un+1(x), which can be bounded independently of ε. To analyze the consistency of the Peaceman and Rachford method, we define the local error en+1 for the time semi-discretization scheme by .
The Spatial Discretization
The piecewise-uniform Shishkin mesh
The finite difference scheme
Error analysis
Next, we will estimate the local error obtained as a result of the discretization of the problem (7.9) by the numerical scheme (7.19) in the y-direction. According to the requirement, we mention the dependence of x ∈ I1,εN in the following expressions, otherwise it is omitted. Now (I) can be estimated in a similar way as in the previous lemma and also we can apply the bounds (7.35) obtained in Lemma 7.4.5 to estimate (II).
Uniform Convergence of the Fully Discrete Scheme
Numerical Results
Since the exact solution of the IBVP (7.48) is known, for each ε the maximum pointwise error is calculated by. The calculated maximum pointwise errors seN,∆tε and the corresponding convergence order pN,∆tε for Example 7.6.1 are shown in Table 7.1 for different values of εand N. From the results in Table 7.1 we see that the calculated ε -uniform errors AND ,∆t decreases monotonically as N increases.
Conclusion
This chapter presents the summary of the contributed results made in this thesis, followed by future goals for possible extensions of the current works. A uniform convergent hybrid numerical scheme is proposed and analyzed for one-dimensional turbulent convection-diffusion parabolic IBVPs with smooth and unsmooth data. In the case of the class of uniquely turbulent parabolic IBVPs with discontinuous convection coefficients, an optimal order of convergence (up to a logarithmic factor) is obtained within the layer regions for the newly proposed hybrid scheme.
Future Scopes
Richardson extrapolation method for singularly perturbed coupled system of convection-diffusion boundary value problems.CMES: Comput. High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with Robin boundary conditions. Appropriate mesh method for singularly perturbed response convection-diffusion problems with boundary and interior layers.
Loglog plot of the maximum point-wise errors
Loglog plot for the spatial order of convergence
Loglog plot of the maximum point-wise errors for Example 3.5.1
Loglog plot of the maximum point-wise errors for Example 3.5.2
Loglog plot of the maximum point-wise errors
Loglog plot of the spatial order of convergence for Example 4.5.1
Loglog plot of the spatial order of convergence for Example .2
Loglog plot of the maximum point-wise errors for Example 5.4.1 over the full
Loglog plot of the maximum point-wise errors for Example 5.4.2 over the full
Loglog plot of the spatial order of convergence for Example 6.5.1
Loglog plot of the maximum point-wise errors for Example 7.6.1
Loglog plot of the spatial order of convergence for Example 7.6.1
