Block Hybrid -Step Backward Differentiation Formulas for Large Stiff Systems

This paper presents a generalized high order block hybrid -step backward differentiation formula (HBDF) for solving stiff systems, including large systems resulting from the semidiscretization parabolic partial differential equations (PDEs). A block scheme in which two off-grid points are specified by the zeros of the second degree Chebyshev polynomial of the first kind is examined for convergence, and stabilities. Numerical simulations that illustrate the accuracy of a Chebyshev based method are given for selected stiff systems and partial differential equations. 1. Introduction We consider the first order differential equation where , , , satisfies a Lipschitz condition, and the eigenvalues of the Jacobian have negative real parts (see [1]). It is well known that the system (1) is better handled by methods with larger stability intervals. In particular, -stable methods are of great importance. However, for very large systems arising from the semidiscretization of parabolic PDEs, -stable methods converge very slowly to the exact solution. Hence, we seek methods which are at least -stable for efficiently solving (1) when the system is very large (see Cash [2]). The development of continuous methods has been the subject of growing interest due to the fact that continuous methods enjoy certain advantages, such as the potential for them to provide defect control (see Enright [3]) as well as having the ability to generate additional methods, which are combined and applied in block form (see Onumanyi et al. [4, 5], Akinfenwa et al. [6], and Jator [7]). The majority of block methods which are due to Shampine and Watts [8], Chartier [9], Rosser [10], and Chu and Hamilton [11] are generally implemented in the predictor-corrector mode. In this paper, we adopt a different approach where the solution is simultaneously provided in each block (see Jator et al. [12], Jator [7, 13]) without the use of predictors from other methods. It is unnecessary to make a function evaluation at the initial part of the new block since at all blocks (with the exception the first block) the first function evaluation is already available from the previous block. The paper is structured as follows. Section 2 states the generalized block HBDF, its derivation from continuous approximation, and how to generate specific members of the scheme. The HBDFs are bundle as main and additional methods, a concept that is due to Brugnano and Trigiante [14]. In Section 3 we study the stability of the schemes with emphasis on a Chebyshev based member. In Section 4 a numerical algorithm for the block