Abstract
We describe the development of a 2-point block backward difference method (2PBBD) for solving system of nonstiff higher-order ordinary differential equations (ODEs) directly. The method computes the approximate solutions at two points simultaneously within an equidistant block. The integration coefficients that are used in the method are obtained only once at the start of the integration. Numerical results are presented to compare the performances of the method developed with 1-point backward difference method (1PBD) and 2-point block divided difference method (2PBDD). The result indicated that, for finer step sizes, this method performs better than the other two methods, that is, 1PBD and 2PBDD.
1. Introduction
In this paper, we consider the system of th order ODEs of the form with in the interval , where For simplicity of discussion and without loss of generality, we consider the single equation where As shown in Figure 1, here the 2-point block method, the interval , is divided into series of blocks with each block containing two points; that is, and is the first block while and is the second block, where solutions to (3) are to be computed.
Previous works on block method for solving (3) directly are given by Milne [1], Rosser [2], Shampine and Watts [3], and Chu and Hamilton [4]. According to Omar [5], both implicit and explicit block Adams methods in their divided difference form are developed for the solution of higher-order ODEs. Majid [6] has derived a code based on the variable step size and order of fully implicit block method to solve nonstiff higher-order ODEs directly. Ibrahim [7] has developed a new block backward differentiation formula method of variable step size for solving first- and second-order ODEs directly. Suleiman et al. [8] have introduced one-point backward difference methods for solving higher-order ODEs. Hence, this motivates us to extend the method to block method in solving nonstiff higher-order ODEs.
2. The Formulation of the Predict-Evaluate-Correct-Evaluate (PECE) Multistep Block Method in Its Backward Difference Form (MSBBD) for Nonstiff Higher-Order ODEs
The code developed will be using the PECE mode with constant stepsize. The predictor and corrector for first and second point will have the following form.
Predictor: where is coefficient for predictor for and .
Corrector: where is coefficient for corrector for and .
We also formulate the corrector in terms of the predictor. Both points and can be written as We derived the formulation for both the predictor and corrector.
3. Derivation for Higher-Order Explicit Integration Coefficients
3.1. For the First Point
The derivation for up to third-order explicit integration coefficients for the first point has been given by Suleiman et al. [8].
3.2. For the Second Point
Integrating (3) once yields Let be the interpolating polynomial which interpolates the values ; then Approximating in (6) with and letting gives or where Define the generating function for the coefficient as follows: Substituting in (14) into gives which leads to Equation (17) can be written as or Hence, the coefficients of are given by Integrating (1) twice yields Substituting with gives The generating function of the coefficient is defined as follows: Substituting (22) into above gives Substituting into (24) yields Equation (25) can be written as or
Hence the coefficients of in relation to coefficients of the previous order are given by By using the same process previously, we note that for integrating times yield and, from (29), we get Integrating times yield or, in the backward difference formulation, given by where The generating function Substituting (35) into above yields As in (30), we now substitute in (37) giving Equation (38) can be written as or Hence the coefficients of in relation to coefficients of the previous order are given by
4. Derivation for Higher-Order Implicit Integration Coefficients
4.1. For the First Point
The derivation for up to third-order implicit integration coefficients for the first point has been given by Suleiman et al. [8].
4.2. For the Second Point
Integrating (3) once yields Let be the interpolating polynomial which interpolates the values ; then As in the previous derivation, we choose Replacing by yields Simplify where Define the generating function for the coefficient as follows: or which leads to For the case , the approximate solution of has the form The coefficients are given by where are the coefficients of the backward difference formulation of (54) which can be represented by Define the generating function of the coefficient as follows: Substituting (54) into above gives Solving (57) with the substitution of (51) produces the relationship By using the same process previously, we note that for integrating times yield Integrating times yield The coefficients are given by where are the coefficients of the backward difference formulation of (62) which can be represented by Define the generating function of the coefficient as follows: Substituting (62) into above gives Solving (65) with the substitution of (59) produces the relationship
5. The Relationship between the Explicit and Implicit Coefficients
5.1. For the First Point
Calculating the integration coefficients directly is time consuming when large numbers of integration are involved. An efficient technique of computing the coefficients is by formulating a recursive relationship between them. With this recursive relationship, we are able to obtain the implicit integration coefficient with minimal effort. The relationship between the explicit and implicit coefficients for the first point is already given by Suleiman et al. [8].
5.2. For the Second Point
For first-order coefficients, It can be written as By substituting into (68), we have
Expanding the equation yields This gives the recursive relationship For second-order coefficient, It can be written as Substituting (70) into the equation above gives or Substituting (25) into (76) gives Expanding the equation yields This gives the recursive relationship By using the same process previously, we note that, for -order coefficient, we have which leads to a recursive relationship For -order coefficient, we have It can be written as Substituting (80) into (83) gives or Substituting into (85) leads to Expanding the equation yields which leads to a recursive relationship Tables 1 and 2 are a few examples of the explicit and implicit integration coefficients.
6. Problem Tested
The problems shown in Table 3 are used to test the performance of the method.
7. Numerical Result
Tables 4, 5, 6, 7, and 8 give the numerical results for problems given in the previous section. The results for the 2PBBD are compared with those of 2PBDD and 1PBD according to Omar [5] and Suleiman et al. [8], respectively. Also given are graphs, where is plotted against and . The following notations are used in the tables: : step size, 2PBBD: 2-point block backward difference method, 2PBDD: 2-point block divided difference method, 1PBD: 1-point backward difference method, NS: total number of steps, MAXE: maximum error, TIME: total execution times (in microsecond).Two sets of scaled graphs were plotted, namely, (i) against and (ii) against . For a particular abscissa, the lowest value of the ordinate is considered to be the more efficient at the abscissa considered. Hence, for the first set of graphs, that is, against , the method 2PBBD is better when , and loses out for value of . For the second set of graphs, as the time increases, the 2PBBD is the method of choice since it is lowest for all five sets of problems (see Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11). It gives us the impression of stability, where the errors grow most slowly compared with the other methods, 2PBDD and 1PBD.
8. Conclusion
Of the 3 methods, 2PBBD is therefore preferred as a general code and should be included as a collection of methods, as a code for parallelization purposes, as an assembly of codes to be tested and studied, and as a code for solving ODEs.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors gratefully acknowledge that this research was supported by Universiti Putra Malaysia, GB-IBT Grant no. GP-IBT/2013/9410100.