The orthogonal collocation method is compared with the collocation method and the advantage of the former is illustrated. Chapter 3 explains the features of the orthogonal collocation method on finite elements and is then used to solve an ODE.
Introduction
Therefore, for a well-conditioned problem, the closer the residual is to zero, the better the approximate solution. The unknown coefficients ci are chosen to force the residual to zero in some average sense over the X domain by requiring that.
Variation on Theme of Weighted Residuals
- Collocation Method
- Sub-domain Method
- Least Squares Method
- Moment Method
- Galerkin Method
The ease of performing integrals with the Dirac delta function is an additional advantage of the collocation method. The weighting functions are chosen as the derivative of the approximation function with respect to the point, i.e.
Numerical Examples
- Two Terms Trial Solution for a Slow Reaction Rate
- Two Terms Trial Solution for a Fast Reaction Rate
- Three Terms Trial Solution for a Fast Reaction Rate
- Different Choice of Collocation Points
The weight function is the same as the test function, so we choose w1(x) = (1−x2). Example 1.3.3 To improve the accuracy of the solution for a fast reaction rate α= 10, we keep three terms in the trial solution.
Introduction
Orthogonal Polynomials
- Useful Properties of Orthogonal Polynomials
- Chebyshev Polynomials of the First Kind T N (x)
- Chebyshev Polynomials of the Second Kind U N (x)
- Legendre Polynomials P N ( x )
The Chebyshev polynomials of the first kind [60] arise as the solution to the Chebyshev differential equation. The Chebyshev polynomials of the first kind are orthogonal with respect to the weight function w(x) = 1/√.
Lagrange Interpolation
Chebyshev Interpolation Nodes
Relationship Between Galerkin and Colloca- tion Methodtion Method
Thus, if the collocation method is used with the collocation points chosen as the roots of the orthogonal polynomial φN(x), then the collocation method will closely approximate the Galerkin method. If φN(x) were not an orthogonal polynomial, then its roots could be complex and could also extend outside the domainX. The collocation method is easy to apply and program, and its accuracy can be compared to the Galerkin method if the collocation points are chosen judiciously [21, 22].
Note that our choice of collocation pointsx1 = 0.25 and x2 = 0.75 in chapter 1 (See section 1.3.4) gave good results because we use the shifted roots of the Chebyshev polynomial of the second kind which falls under the family of orthogonal polynomials has.
Numerical Examples
One Point Collocation for Example
The solution representing the concentration of the chemical becomes negative for α = 10 and since this cannot happen in reality, we conclude that more collocation points are needed to produce a higher order approximation.
Generalization of Example 2.5.1
We first solve the problem by choosing the collocation points xjc, j N as the shifted roots of the Chebyshev polynomial of the first kind, TN−1. The plots of the solutions and errors for orderN = 3, 4 and 5 are given in figures 2.5 and 2.6 respectively. We notice the gradual decrease in the total error as we increase the order of the polynomial.
Second, we choose the collocation points as shifted roots of the Čebyš polynomial of the second kind, UN−1. The solution and error graphs for orders N = 3, 4, and 5 are given in Figures 2.8 and 2.9, respectively. Third, we choose the collocation points as the shifted roots of the Legendre polynomial, PN−1.
Comparing the total error from Table 2.3, we observe that the total error is least for UN−1. Note that in the previous example we moved the roots of the orthogonal polynomials from [−1, 1] to the domain of the problem, namely [0, 1].
Two Points Collocation for Example 2.5.2
We choose the first and the last interpolation point to coincide with the left and right boundary points, respectively, that is, x1 =−1 and x4 = 1, and we choose the remaining interpolation points as the zeros of the Chebyshev polynomial T2, that is, x2. The two collocation points x2c and x3c are chosen as the roots of T2 and therefore coincide with the internal interpolation points x2 and x3 respectively. Therefore, we have four cubic Lagrange polynomials given by 2.63) Substituting the approximate solution (2.59) into the differential equation (2.57) and forming the remainder gives.
The solution and error graphs are shown in Figures 2.12 and 2.13, respectively. Note from Table 2.4 that the numerical solution for various values of x does not agree closely with the analytical solution. Furthermore, the error is high (see Figure 2.13), so we require more collocation points to get a better approximation.
Generalization of Example 2.5.2
In Figure 2.14, we notice that as we increase the order of the polynomial, the approximate solution gradually converges to the exact solution, and the error decreases at the same time, as shown in Figure 2.15 and Table 2.5. However, notice from Figures 2.14 and 2.15 that the solution obtained from the order of N = 10 is not a very good approximation. This is one of the limitations of higher order polynomials and we will have to consider other means if we want to get more accurate results.
Second, if we choose the roots of the Chebyshev polynomial of the second kind, UN−1 as the collocation points, then we obtain the plots of the solutions and the corresponding errors for orders N as shown in Figures 2.17 and 2.18, respectively. Note from Table 2.5 that the Legendre polynomial gives better results for higher orders (orders 10 and higher). The error levels off at order N = 17 while the error of the Chebyshev polynomials of the first and second kind levels off at order N = 20 with value 1.4216e−005.
From the error graphs in Figures and 2.20, we notice that the approximate solution is much better in the region x ≥ 2 where the solution flattens compared to the region near the hump or peak.
Introduction
The accuracy, compatibility and easy adaptability of the finite element method makes it suitable to be applied together with the collocation method. In the case of orthogonal colocalization, when the reaction rate is very high compared to the diffusion rate, the concentration profile of the chemical is very sharp, so a high-order polynomial will be required to achieve reasonable accuracy.
Methodology
Distribution of Elements
Figuring out how to distribute the elements within any given domain [a, b] is a complex step that was quite a challenge with the finite element method. He suggested that better results can be obtained in the case of unequal element spacing than with equal element spacing. Due to the uneven spacing of the elements, we will place smaller elements in areas where the solution is steep, and larger ones where the solution changes less.
Numerical Example
Two Finite Elements for Example
Since we require two collocation points per element, they will be chosen as the roots of an orthogonal polynomial of order two, namely uc2 and u3c. Substitution of the approximate solution in equation (3.5) with differential equation (3.6) gives the residual in the ith element. We satisfy the residual equation for each element at the collocation points u2c and u3c to obtain. 3.10).
Since we have a total of eight unknowns, we need two additional equations for a unique solution. 3 are chosen as the roots of the Legendre polynomial, P2 since they gave good results in chapter 2. Comparing the total error for y2 and y¯2 from Table 3.1, we note a reduction of 17.24% for the latter case compared to the former.
Generalization of Example 3.3.1
A set of discrete numerical results for y3 and y¯3 is summarized in Table 3.2, from which we deduce that the error is reduced by 28.92% when we use oddly distributed elements. Therefore, it is obvious from our analysis that properly chosen odd element spacing gives superior results as suggested by Finlayson [22]. As we increase the number of elements, the approximate solution tends to converge to the exact solution as shown in Figure 3.7fory6 and ¯y6.
The advantage of this is that the size of the matrix is reduced, as well as the numerical effort for the solution of the linear system.
Introduction
Numerical Example
Exit Solution
However, the output solution given in equation (4.4) converges too slowly to be very practical for large P and/or small t. We obtain the required output solution by setting x = 1 in equation (4.5) and simplifying it, we have As in the case of OCFE applied to ODE, the domain [0, 1] is divided into Ne elements.
We obtain the approximate solution in the element i using a cubic Lagrange polynomial as in Chapters 3, but noting that the approximate solution is a function of the second and t, we write Substituting the approximate solution in equation (4.8) into the partial differential equation (4.9) gives the remainder in the ith element, that is. There are a total of 4Ne of coupled DAEs which we solve using MATLAB with the ode15s subroutine [64].
For convenience, the approximate solution from the OCFE method using Ne equal elements will be denoted by yNe. As we increase the number of elements, the approximate output solution yi quickly converges to the output solution.
General Solution
Limiting Cases
Conclusion
Conclusion
The method of OCFE, which combines the features of the orthogonal collocation method with the finite element method, was found to be numerically more stable and reliable than the orthogonal collocation method, especially for problems with steep gradients (see Example 3.3.1). For the orthogonal collocation method that uses an orderN approach, we need to evaluate (N −1)×(N + 1) entries of the submatrix defined by rows 2 through N of A (see equation (2.71)). Most researchers of the orthogonal collocation method [21, 36] have solved initial and boundary value problems using the zeros of the Jacobi polynomials.
Therefore, one could choose the collocation points as the roots of any of the orthogonal polynomials, except in cases where studies have shown otherwise. In the case of the OCFE method, we restricted the trial solution to lower order polynomials (order 3 to be specific) and the results are comparable to those given in previous work (see [4]). Thus, a possible area for future research is the application of the OCFE method to higher dimensional problems.
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