Circumventing Ill-Conditioning Arising from Using Linear Multistep Methods in Approximating the Solution of Initial Value Problems

Abstract

When finding numerical solutions to stiff and nonstiff initial value problems using linear multistep methods, ill-conditioned systems are often encountered. In this paper, we demonstrate how this ill-conditioning can be circumvented without iterative refinement or preconditioning, by carefully choosing the grid point used in deriving the discrete scheme from the continuous formulation. Results of numerical experiments show that the new scheme perform very well when compared with the exact solution and results from an earlier scheme.

Comparing the Performance of Two Butcher-Based Block Hybrid Algorithms for the Solution of Initial Value Problems

“…In this work, we extend the idea in [7] to numerical examples of two, three, four and six-dimensional systems of differential equations to confirm the earlier claim that y n+ 5 2 is better well-conditioned than those of y n+ 7…”

“…The two-step Butcher's hybrid scheme has been derived by several authors [3][4][5][6][7][8] using different off-grid points. The two-step Butcher's block hybrid method is a linear multi-step of order five for the numerical solution of one-and high-dimensional systems of ordinary differential equations which is the crux of the matter in this article.…”

“…In an earlier work, we showed that, for one-dimensional stiff and non-stiff, linear and nonlinear differential equations, the performance of the Butcher-based block hybrid method obtained at the points {1, 3 2 , 7 4 , 2} with those obtained at {1, 3 2 , 2, 5 2 } were at par, though the latter was better conditioned than the former. In order to avoid abuse of notation, for the purpose of this study, we will refer to y n+ 7 4 as the block hybrid method obtained at the following combination of grid and off-grid points: {1, 3 2 , 7 4 , 2} and y n+ 5 2 the block hybrid method obtained at the latter points {1, 3 2 , 2, 5 2 } in line with [7]. In fact, this work is an extension of the works of Akinola et al [7], where both methods were shown to be of order five and convergent.…”

“…In order to avoid abuse of notation, for the purpose of this study, we will refer to y n+ 7 4 as the block hybrid method obtained at the following combination of grid and off-grid points: {1, 3 2 , 7 4 , 2} and y n+ 5 2 the block hybrid method obtained at the latter points {1, 3 2 , 2, 5 2 } in line with [7]. In fact, this work is an extension of the works of Akinola et al [7], where both methods were shown to be of order five and convergent. Nevertheless, we did not compare their regions of absolute stability nor their error constants because the goal in that study was to provide a brief solution to Shampine's [1] claim about the ill-conditioning of matrices obtained from linear multi-step methods (see also [9][10][11][12][13][14][15][16]) and solving first order differential equations.…”

“…In addition to our discussion in the second to the last paragraph above, the focus in [7] was to discuss with numerical examples how ill-conditioning arising from using linear multi-step methods in approximating the solution of initial value problems could be reduced, albeit for one-dimensional ODEs, which is also an improvement from the works in [8]. Similarly, Adee and Atabo [17] presented a two point linear multi-step method for solving similar problems as the ones considered in [7] and the present work, except that their method is non starting and they used a fourth order Runge-Kutta method in obtaining the starting values. In this work, our method is self-starting and does not rely on other methods to start (see also [18][19][20][21][22][23][24][25].…”