Natesan Srinivasan, 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. Udai Sankar of the Department of Mathematics, IIT Guwahati for their help in some related official matters.
Brief Background
The author gave a new derivation of the decomposition of the solution into the regular and the singular components. 14, 15] deal with the numerical approximation of the solution of 1D parabolic single perturbed problems of reaction diffusion and convection diffusion types, respectively.
Objective and Motivation
Objective and Motivation to generate a grid that is adapted to the solution's functions. Sufficient conditions are derived on the monitor function which are independent of the perturbation parameter and guarantee uniform convergence in the discrete maximum norm.
Some Notations and Terminology
- Ideas of mesh-construction
- The Shishkin mesh
- The Bakhvalov mesh
- Adaptive spatial meshes via equidistribution
Here we report the piecewise uniform Shishkin mesh, a frequently studied simple mesh, for the spatial discretization of the domain Ω for the problem (1.3.1). From these considerations, a typical convergence result for simple convolution of the problem is (1.3.1) on the Shishkin mesh.
Model Problems
- Singularly perturbed parabolic reaction-diffusion problem
- Singularly perturbed parabolic convection-diffusion problem
- Semilinear singular perturbation parabolic problem
- Singularly perturbed delay parabolic reaction-diffusion problem . 18
The method converges second order in space and first order in time, regardless of the singular perturbation parameter. The method converges second order in space and first order in time, regardless of the singular perturbation parameter.
Analytical Behavior of the Solution
Assume that the coefficients of the parabolic PDE, and the initial and boundary conditions given in (2.1.1) are sufficiently smooth, and satisfy the necessary compatibility conditions as detailed in (2.1.2) and (2.1.3). We will decompose the solution u as u = v +w, where v, w are the regular and the singular components, respectively.
The Numerical Solution
Finite difference scheme
Numerical solution We discretize the PDE (2.1.1) by means of the back-Euler scheme for the time derivative and the central difference scheme for the spatial derivative. After rearranging the terms in (2.3.1), we obtain the following form of the difference scheme, forn = 0.1,.
Adaptive spatial mesh via equidistribution
To determine the value of the monitor function (2.3.5), we need to know the approximate value of the singular component w(x, t). One-dimensional version of the monitor function (2.3.5) given by [6] impressed us to take. 2.3.6) The choice of this α will help to evenly distribute the number of mesh points inside and outside the boundary layer region.
Numerical algorithm
Error Analysis
Considering the IBVP (2.1.1) and the difference scheme (2.3.1), the difference scheme (without initial and boundary conditions) can be written as Problem (2.1.1) shows the correct boundary layers and the same applies to equation (2.4.3), the resulting error bound derived in [7].
Semilinear Parabolic Problem
Numerical Results
The calculated maximum pointwise errors in the normalized flux rN,∆tε and the corresponding convergence rate qεN,∆t for example 2.6.1 are given in Table 2.3 and Table 2.4, respectively. Numerical results Table 2.6: Convergence rate of the solution PεN,∆t for example 2.6.2 using an equilibrium distribution network.
Conclusions
The non-uniform meshes are obtained by an equal distribution of a positive monitor function, which depends on the second-order spatial derivative of the single component of the solution. Particularly disturbed parabolic convection-diffusion problems of the form (3.1.1) arise in various branches of science and engineering.
The Analytic Solution
The Numerical Approximation
- Semidiscretization
- Adaptive spatial meshes via equidistribution
- Finite difference scheme
- Numerical algorithm
To calculate the value of the monitor function M(u(x, T0), x) at the internal node of the spatial grid, Mi, we assume without loss of generality that w(xi, T0) =WiS, where S∆t =T0 ,. This is equivalent to approximating the monitor function by the piecewise constant function. To determine the value of the monitor function (3.3.7), we need to know the approximate value of the singular component w(x, t).
Error Analysis
Decomposition of numerical solution
The local truncation error of the regular component at node (xi, tn+1) will be given by. Using Peano's kernel theorem for any value of n, the regular component truncation error reduces to. The local truncation error of the singular component at node (xi, tn+1), is given by τi,n+1Wc = I+ ∆tLN.
Uniform convergence of the fully discrete scheme
Error Analysis Now using Υn+1j = CN(1+Si), as the barrier function and the discrete maximum principle (Lemma 3.4.1), we get. Let bun+1 and {Ubn+1} be the solutions corresponding to the semidiscrete discretization (3.3.3) and the discrete solution (3.4.1), respectively. Using the error bounds for the regular and singular components from Lemmas 3.4.2 and 3.4.4 in the inequality (3.4.5), we obtain the required estimate. 3.4.12) This limit is necessary to prove the uniform convergence of the fully discrete scheme.
Semilinear Parabolic Problem
Numerical Results
The calculated maximum pointwise errors in the normalized flux rN,∆tε and the corresponding convergence rate qεN,∆t for example 3.6.1 are given in Table 3.3 and Table 3.4. Numerical results Table 3.11: Maximum pointwise error of the normalized flux RN,∆tε for example 3.6.2 using an equilibrium distribution network. Numerical results Table 3.13: Maximum pointwise error of the solution ˜eN,∆tε for example 3.6.3 using an equidistribution network.
Conclusions
Error estimates are derived for the numerical scheme, which are independent of the diffusion parameter. To solve these problems, we use a modified back-Euler scheme for the time derivative with a uniform grid and a central difference scheme for the spatial derivative on nonuniform layer-matched grids at each time level. Nonuniform grids are obtained by uniformly distributing a positive monitor function, which includes the second-order spatial derivative of the singular component of the solution.
The Numerical Solution
Finite difference scheme
We discretize equation (4.1.1) using the modified posterior difference for the time derivative and the central difference for the space derivative. After rearranging the terms in (4.2.1), we get the following form of the difference scheme: forn= 0,1,. To determine the value of the monitor function (4.2.5), we need to know the approximate value of the singular component w(x, t).
Adaptive spatial grids via equidistribution
Numerical algorithm
Error Analysis
Using the assumption that b(x) is often independent, the definition (4.3.2) implies that δt∗φ satisfies. 4.3.6) To analyze the above sequence of two-point boundary value problems (4.3.6), observe that the right-hand side of the above equation can be written as The above step is just to use the linear interpolation, you can also use higher order interpolation such as cubic spline for higher order convergence speed. Using the fact that the interpolation error is of order O(N−2), one can obtain the following inequality.
Numerical Results
Conclusions
Conclusions Table 4.1: Maximum pointwise error of the solution eN,∆tε for example 4.4.1 using an equilibrium distribution network. Conclusions the grids properly, and the spatial derivative is replaced by the central difference scheme. The non-uniform rasters are obtained by equally dividing a positive monitor function, which is a linear combination of a constant and the second-order spatial derivative of the single component of the solution at each time level.
The Numerical Solution
Finite difference scheme
After rearranging the terms in (5.2.1), we get the following form of the difference scheme: forn = 0.1,. To determine the value of the monitor function (5.2.6), we need to know the approximate value of the singular component w(x, t). To calculate the numerical value of Win for w(xi, tn), we use the numerical approximate value of Vin for v(xni, tn) from the following inverse relation: forn = 0.1,.
Adaptive spatial grids via equidistribution
Error Analysis
Decomposition of the numerical solution
Error Analysis,By using Peano's core theorem for any value ofn,the truncation error of the regular component is reduced to,. We prove this lemma by separating the region into the regular region and the layer region, i.e. i= 0.1. where we used Lemma 5.3.3 and the bounds of the single component w. As before, the truncation error of the single component can be bounded as follows.
Uniform convergence of the fully discrete scheme
By using the error bounds of the regular and the singular components from Lemmas 5.3.2 and 5.3.4 in the inequality (5.3.5), we obtain the required estimate. 5.3.12) This bound is required to prove the uniform convergence of the fully discrete scheme.
Semilinear Parabolic Problem
Numerical Results
The calculated maximum point errors of the normalized flow rN,∆tε and the corresponding degree of convergence qεN,∆t for case 5.5.1 are given in table 5.3 and table 5.4. Numerical results Table 5.5: The maximum point error of the solution EεN,∆t for case 5.5.2 using an equal distribution network. Numerical results Table 5.12: Convergence rate of the normalized current qεN,∆t for example 5.5.3 using an equal distribution network.
Conclusions
Robust numerical scheme for singularly perturbed parabolic initial-boundary-value parabolic problems on equal networks. The domain is discretized with a uniform grid in the time direction and a non-uniform grid obtained by uniformly distributing a monitoring function for the spatial variable. The numerical scheme consists of the implicit-Euler scheme for the time derivative and the classical central difference scheme for the spatial derivative.
Introduction
The main goal of this chapter is to provide an ε-uniform numerical method for the IBVP (6.1.1) on a flexible mesh. In this method, the time derivative is replaced by a backward Euler scheme, and the spatial derivative is replaced by a central difference scheme. We then perform an error analysis for the upwind scheme in Section 6.4 and prove the main theoretical result, viz. ε-Unit optimal error bounds.
Analytic Solution
Therefore, in order to obtain stronger estimates of the solution u(x, t) and its partial derivatives, we decompose the solution u(x, t) into regular and singular components. With the above definition of v0 and v1, we can define the regular component v and the singular component w as follows. The following theorem gives the bounds of the regular component v and the singular component w and its partial derivatives, which play a key role in the error analysis in Section 6.4.
The Numerical Solution
Finite difference scheme
To determine the value of the monitor function (6.3.5), we need to know the approximate value of the singular component w(x, t). To calculate the numerical valueWinofw(xi, tn), we use the numerical approximate value Vin of v(xi, tn) from the following recurrence relation: for n Then the value ofWin will be calculated from Win=Uin−Vin.
Adaptive spatial grids via equidistribution
The numerical solution of the monitor function at the ith internal node of the spatial grid, Mi, is assumed to be w(xi, T0) =WiS, where S∆t =T0,. For a truly adaptive algorithm, the monitor function must be approximated from a numerical solution. In the numerical algorithm given in Section 2.3.3, we use the discrete form of the control function given in (6.3.6).
Error Analysis
Therefore, the truncation error of the above scheme can be written in the following way, as given in [49] and [17]. To analyze the above set of two-point boundary value problems (6.4.7), note that the right-hand side of the above equation can be written as By fixing this, the lateral part of the above inequality can be viewed as the truncation error of the reaction-diffusion two-point boundary value problem as in [7], corresponding to the left layer part.
Numerical Results
The maximum calculated error at point N,∆tε and the corresponding rate of convergence pN,∆tε for example 6.5.1 are given in Table 6.1 and Table 6.2, respectively. The maximum calculated pointwise errors in the normalized flux rN,∆tε and the corresponding rate of convergence qN,∆tε for Example 6.5.1 are given in Table 6.3 and Table 6.4. Numerical results Table 6.3: Maximum error in normalized flux point rN,∆tε for example 6.5.1 using equal distribution grid.
Conclusions
The proposed numerical scheme is of first order in the temporal variable and second order in the spatial variable, that is, O(∆t+N−2). Error estimates are derived for the numerical scheme, which are independent of the diffusion parameter ε. We partition the domain using a piecewise uniform Shishkin mesh in the spatial direction and uniform mesh in the temporal direction.
The Analytic Solution
The characteristics of (7.2.1) are the vertical linesx= constant, which implies that any boundary layer appearing in the solution is of the parabolic type. We decompose the exact solution u(x, t) of IBVP (7.1.1) into the regular and singular components as. We take further decomposition of the regular component v for any prescribed order k by assuming the necessary compatibility condition as. k, are the solutions to the following first-order problems. and finally satisfies the function vk+1.
The Numerical Solution
Finite difference scheme
Since the problem (7.1.1) has only one regular layer at x = 1, we divide the domain Ω into two subdomains [0,1−ρ] and [1−ρ,1] to define the piecewise uniform mesh. and then divide each subdomain into N/2 equal intervals with the points. Here we define the discretization of domains in a systematic way to make our presentation clear. Error analysis In the same way, we define the right and left boundary points by ΓNl = GN,∆t ∩Γl and.
Error Analysis
Error Analysis Therefore, the clipping error of the above scheme can be written as follows, as reported in [49] and [17]. The two-point boundary value problem (7.4.3) implies that δtφ satisfies. 7.4.9) To analyze the above sequence of two-point boundary value problems (7.4.8), note that the right-hand side of the equation can be written as Numerical results By fixing t, the side part of the above inequality can be seen as a truncation error.
Numerical Results
The calculated maximum point-wise errors eN,∆tε , eN,∆t; and the corresponding rate of convergence spN,∆tε, pN,∆t for example 7.5.1 is given in table 7.1. From Table 7.2 we observe that the convergence of the error corresponding to the regular part is close to first order. From the results given in the tables, we see the monotonically decreasing behavior of the calculated ε-uniform errors.
Conclusions
Moreover, the errors in the normalized flux of the exact and the computational solutions are presented in the numerical results section. Numerical results of the errors are plotted in the log-log scale to show the convergence rate. In addition, numerical results of the errors are plotted in the log-log scale to show the convergence rate.
Future Scopes
Convergence analysis of finite-difference approximations on uniformly distributed grids to the singularly perturbed boundary value problem. Numerical solution of a singularly perturbed two-point boundary value problem using equal distribution: convergence analysis. On the HP finite element method for one-dimensional singularly perturbed convection-diffusion problems.
Loglog plot for Example 2.6.1
Numerical solution of Example 2.6.2 for N = 32 and ∆t = 1/32
Loglog plot for Example 2.6.2
Loglog plot for Example 3.6.1
Numerical solution of Example 3.6.2 for N = 64 and ∆t = 1/64
Loglog plot for Example 3.6.2
Solution and error plots for Example 4
Loglog plot for Example 5.5.2
Numerical solution at various time levels of Example 5. for N = 16
Loglog plot for Example 6.5.1
Numerical solution of Example 6.5.2 for N = 64 and ∆t = 0.01
Numerical solution of Example 7.5.2 for N = 64 and ∆t = 0.1
