Then, for the same problem, we apply the Richardson extrapolation technique to improve the accuracy of the implicit upwind scheme on the piecewise uniform Shishkin grid in the spatial direction and the uniform grid in the temporal direction. Next, depending on the sign of the discontinuous convection coefficients in the domain, we study the numerical solutions of different types of single-perturbed two-dimensional elliptical convection-diffusion BVPs.
Brief Background
The differential equation becomes particularly perturbed when the magnitude of the highest-order derivative is dominated by lower-order terms and is often attributed to a small parameter that is multiplied by the highest-order derivative. Usually, these lower-order differential equations do not satisfy all the boundary or initial conditions.
Objective and Motivation
Natividad and Stynes [67] used this technique to solve singularly perturbed convection-diffusion two-point BVPs. So far we have discussed singularly perturbed convection and diffusion problems where the solutions exhibit only boundary layers.
Some Notations and Terminologies
Further discussion on the choice of the norm can be found in the book by Miller et al. To measure the performance and the robustness of the numerical method, let us introduce the concept of ε-uniform convergence.
Model Problems
- Singularly perturbed 1D parabolic convection-diffusion problem . 9
- Singularly perturbed 1D parabolic convection-diffusion problem
- Singularly perturbed 2D elliptic convection-diffusion problem with
- Singularly perturbed 2D parabolic convection-diffusion problem
The convection coefficient a and the source term f have discontinuity at x = ξ and satisfy the following property. The convection coefficient a and the source term f have discontinuity at x = ξ, and the jump is finite.
General Outline of the Thesis
Next, to discretize the spatial derivatives of the resulting semidiscrete time problem, we apply the hybrid scheme, which is a combination of the central difference scheme (in the inner region) and the middle upwind scheme (in the outer region) to the piecewise uniform Shishkin schedule. gauze. Then, to discretize the spatial derivatives of the resulting semidiscrete time problem, we apply the hybrid scheme to the piecewise uniform Shishkin mesh.
Analysis of the Continuous Problem
In Section 2.5, we discretize the spatial domain using the piecewise-uniform Shishkin mesh and then, to approach the semidiscrete problem, we introduce the hybrid finite difference scheme, which is a combination of an inner region central difference scheme and a middle windward schedule in the outer area. region. This is easy to demonstrate from the preconditions given in (2.1.1) and by differentiating (2.2.1) according to tox.
The Time Semidiscretization
Therefore, again by fixing t∈[0, T] and following the idea of Kellogg and Tsan [42], one can derive the necessary estimate foruxx. 2.3.2) One can conclude the consistency result for the semidiscrete scheme by using the following lemmas.
Asymptotic Behavior of the Solution of the Semidiscrete Problem
Under the assumptions of Lemma 2.3.2, the global error En of the scheme (2.3.1) satisfies the following bound.
Discretization of the Spatial Domain and the Derivatives
2.5.8) The above tridiagonal system of linear algebraic equations can be solved by any existing solver.
Convergence Analysis
From the estimates given in Lemma 2.4.2 and the local truncation error expression (2.6.13), we can obtain If ν, ∆t is taken such that0< ν <1 and N−ν ≤C∆t, there exists a positive constant C that is independent of ε, N, such that.
Numerical Results
The maximum pointwise errors and the corresponding numerical convergence order are shown in Table 2.1 for p = 3. From Table 2.3 it can be seen that the convergence order in the outer region is two, while in the inner region it is close to two.
Conclusion
Here, we apply the Richardson extrapolation technique to the model problem to improve the accuracy of an upwind finite difference scheme on a piecewise uniform Shishkin grid. Here we use the well-known post-processing technique of the Richardson extrapolation method to improve the accuracy of the numerical solution (3.1.1).
Analysis of the Continuous Problem
The smooth and the singular components v and w, defined in (3.2.4) and (3.2.5) respectively, satisfy the following bounds. One can easily obtain the required bound for the singular component w using the following barrier functions.
Construction of the Numerical Method
The numerical solution
The bounds for the higher-order derivatives of w can be obtained by following the ideas given in [61] and [53]. The following theorem states that the numerical scheme given in (3.3.1) converges ε-uniformly on the Shishkin mesh GMN with near first-order accuracy.
Motivation for applying Richardson extrapolation technique
The main goal of this chapter is to obtain an ε-uniform convergent second-order numerical solution of IBVP (3.1.1) using the Richardson extrapolation method. To obtain an estimate of the nodal error, similar to the federated solution, the numerical solution UN, M on the GMN grid is decomposed into a sum.
Error Analysis
Error estimate for the smooth component
We then use (3.3.4) to determine the error bounded to the smooth component of the extrapolated solution. The error associated with the smooth component of the extrapolated solution corresponds to the following limit.
Error estimate for the singular component
The error associated with the singular component of the extrapolated solution satisfies the following bound. Now, we derive the error estimate for the singular component of the extrapolated solution in the (inner) boundary layer region [0, σ).
Error estimate for the extrapolated solution
Numerical Results
The maximum point errors and the corresponding numerical order of convergence (before and after the Richardson extrapolation) for Examples 3.5.1 and 3.5.2 are shown in Tables 3.1 and 3.2. To show the influence of the parameter p in the solution of the PDE, we have calculated the maximum pointwise errors and the order of convergence for different values of p by fixing ε = 2−15.
Conclusion
This figure again ensures that the order of convergence of the upwind scheme has been improved from O(N−1ln2N + ∆t) to O(N−2ln2N + ∆t2) after applying the Richardson extrapolation technique, which validates the theoretical limit that is given in Theorem 3.4.14. The rest of the chapter is organized as follows: In Section 4.2, we describe the semidiscrete problem by introducing an alternating direction scheme and study the uniform convergence of the semidiscrete scheme.
Time Semidiscretization
Before proceeding to prove the consistency result of the semidiscrete scheme (4.2.9), we prove that the partial derivatives of u with respect to t are uniformly bounded. Under the assumptions of Lemma 4.2.2, the global error En in the scheme (4.2.8) satisfies the following limit.
The Fully Discrete Scheme
Let Lx,εN (and correspondingly Ly,εN) be the discretization of the differential operator Lx,ε. Ly,ε), obtained by applying the upwind finite difference scheme to IxN (IyN), for each y∈IyN. If A is the tridiagonal matrix associated with the finite difference method (4.3.1), then one can conclude from (4.3.4) that −A is an M-matrix.
Convergence Analysis
Local truncation error
The local truncation error associated with (4.3.1), at an internal mesh point xi is given by. For someν,∆t such that0< ν <1 and N−ν ≤C∆t, there exists a positive constant C which is independent of ε, N, such that.
Numerical Results
In Table 4.4 we present the maximum nodal error and the corresponding order of convergence for case 4.5.2 with ε = 2−30 for different values of p, q. To confirm the first-order convergence in time, we present the maximum nodal error and the corresponding numerical order of convergence for Example 4.5.1 for p= 1, q= 2 and different values of ε in Table 4.2.
Conclusion
The rest of the chapter is organized as follows: Section 5.2 contains some limits for the solution of the problem (5.1.1). In the same section, the stability of the numerical scheme and the decomposition of the discrete solution are also discussed.
Analysis of the Continuous Problem
The Discrete Problem
Stability
The following theorem ensures that the discrete operator LeNε satisfies the discrete minimum principle and that the proposed scheme is therefore uniformly stable.
Decomposition of the discrete solution
Error Analysis
Before starting the error analysis for the singular component of the solution, we define the following mesh function. with the usual convention that SN = 1) kuehj =hN+1−j and is a positive constant. First, we set the error estimate in the domain ΩN ∩[σ, ξ) and then follow the same argument to find the associated error in the domain ΩN ∩[ξ+σ,1).
Numerical Results
To compare the proposed scheme and the upwind scheme [26], we have applied the upwind scheme to model example 5.5.1, and the maximum nodal error and the corresponding order of convergence are given in Table 5.2. From the model problem and the bounds of the derivatives, one can expect the solution of the model problem to have a boundary layer near x = 0 and a weak inner layer near x = ξ.
Conclusion
First, to discretize the domain, we use the piecewise uniform Shishkin mesh in the spatial direction and the uniform mesh in the temporal direction. In Section 6.3, we consider the uniform mesh and the piecewise uniform Shishkin mesh to discretize the time domain and the spatial domain, respectively.
Properties of the Analytical Solution
We also describe the numerical scheme to approximate the IBVP (6.1.1) and the stability of the numerical scheme in the same section. In section 6.5 some numerical results are presented in the form of tables to compare the performance between the windward scheme and the hybrid scheme.
The Numerical Solution
Stability
The following theorem ensures that the discrete operator Let,Nε satisfies the discrete minimum principle and therefore the proposed scheme is uniformly stable. Thus, by assuming (6.3.13) we can conclude that A is an M matrix and B ≥ 0 and thus the discrete operator Let,Nε satisfies the discrete minimum principle.
Decomposition of the discrete solution
Then the discrete operator eLt,Nε satisfies the discrete minimum principle, i.e. if {Zin} are network functions satisfying Zin ≤0, on GMN \GMN and Naj,Nε Zin≥0, in GMN, then Zin≤0 in GMN.
Error Analysis
Numerical Results
We define the maximum knot error and the corresponding convergence order for each ε in the same way as we define in Chapter 2. The maximum knot error and the corresponding convergence order for Examples 6.5.1 is shown in Table 6.1, for ξ = 1 /2 and different values of ε.
Conclusion
Type-1: Weak interior layer & boundary layers
We assume that b is sufficiently smooth in the domain and f satisfies sufficient compatibility conditions at the vertices of the domain and 0 < ε ≪ 1. Due to the discontinuity of a (positive throughout the domain) and f on Γ±x, the solution of (7.1.1) exhibits a weak inner layer along linex=ξ and boundary layer along the line x = 0.
Type-2: Strong interior layer & boundary layers
Type-3: Strong interior layers
But the nature of the inner layer depends on the sign of the convection coefficients in the domain. The mesh structures depend on the location of the layers for the three different two-dimensional elliptical BVPs.
Discretization of the Domain
For Type-1
Table 7.1 is given to better understand the position of the layers of three different types of BVP. The rest of the chapter is organized as follows: In Chapter 7.2, we discretize the spatial domain using a piecewise uniform Šiškin grid.
For Type-2
Thus, the discretized domains ΩNx and ΩNy look like ΩNx ={xi}N0 , with xN/2 =ξ and ΩNy ={yj}N0 .Therefore, we denote the discrete domain asDN, i.e. So, the discritze domains ΩNx and ΩNy look like ΩNx ={xi}N0 , with xN/2 =ξ and ΩNy ={yj}N0 .Therefore we denote the discrete domain asDN, i.e.
For Type-3
Finite Difference Scheme
For Type-1
For Type-2
For Type-3
Numerical results
Example for Type-1
Example for Type-2
Example for Type-3
The calculated maximum nodal error and corresponding order of convergence for Examples and 7.4.3 are presented in Tables 7.2, 7.3 and 7.4, respectively, for various values of ε. To reveal the numerical order of convergence for different values of ε, we have plotted the maximum pointwise errors (in loglog scale) in Figures 7.5, 7.7 and 7.9 for Examples and 7.4.3 respectively, which again confirms the near first-order convergence of the proposed numerical scheme.
Conclusion
We then use the upwind difference scheme on the Shishkin grid to discretize the spatial derivatives. Then, to discretize the resulting set of ordinary differential equations, we apply the upwind finite difference scheme to the piecewise uniform Shishkin mesh in the spatial direction.
The Time Semidiscretization
Section 8.3 contains the discretization of the spatial domain using a piecewise uniform Shishkin mesh. In order to analyze the convergence of the semi-discrete scheme (8.2.3), we define locally erroren+1 as
Discretization of the Spatial Domain
The Fully Discrete Scheme
If A is the tridiagonal matrix associated with the finite difference method (8.4.1), then from (8.4.6) one can conclude that A is an M-matrix. Correspondingly, (I + ∆tLNy,ε) also satisfies the discrete maximum principle, and therefore the method is uniformly stable in the highest norm.
Convergence Analysis
Bounds for the solution of the semidiscrete problem
We decompose (x) as z(x) =b bv(x)+w(x), where respectively the smooth and the inner layer components of z can be obtained. For the error analysis, we need to know the boundary of the exact solution bz(x) of the one-dimensional two-point BVP (8.5.1) and its spatial derivatives.
Decomposition of the discrete solution
Error analysis
We first determine the error estimate in the domain ΩNx ∩(0, ξ−σ1,x] and then follow the same argument to find the error bounded in the domain ΩN ∩[ξ+σ2,x,1). Specifically, the proof of the theorem in the outer region is discussed in Example 1, and the proof in the inner region is in Example 2.
Numerical Results
From the results given in Tables 8.1, we can observe that for fixed ε, the calculated maximum nodal errors monotonically decrease as N increases. As a result, even though the perturbation parameter ε is very small, the proposed method still gives acceptable numerical results.
Conclusion
Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient. Mathematics. Applied mesh method for singly perturbed reaction-convection-diffusion problems with boundary and interior layers.
