A family of r-points 1-block implicit methods with optimized region of stability for stiff initial value problems in ordinary differential equations

Abstract

In this paper, we proposed a family of r-points 1-block implicit methods with optimized region of stability. This family of methods is derived with Mathematical 10.4 software and the stability is investigated using boundary locus techniques. The block methods are consistence, zero stable, and Astable and satisfy other stability requirements which finds them suitable for stiff problems in ODEs. Numerical experiments are presented and results are compared with other block methods and exact solutions of some stiff ordinary differential equations. The methods have been found to show competitiveness with other numerical methods.

High Order Continuous Extended Linear Multistep Methods for Approximating System of ODEs

“…Some other works related to multi-derivative schemes include: [7][8][9][10]. The use of hybrid schemes to bypass the Dahlquist Barrier theorem for LMMs has also been considered; see [11][12][13][14][15][16][17][18][19]. However, it was noted in Gupta [20] that the algorithm procedure for the hybrid scheme is more computationally intensive due to the occurrence of an off-step function in the method, which requires more predictors during implementation.…”