Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations

Abstract

This paper aims to select the best value of the parameter ρ from a general set of linear multistep formulae which have the potential for efficient implementation. The ρ -Diagonally Implicit Block Backward Differentiation Formula ( ρ -DIBBDF) was proposed to approximate the solution for stiff Ordinary Differential Equations (ODEs) to achieve the research objective. The selection of ρ for optimal stability properties in terms of zero stability, absolute stability, error constant and convergence are discussed. In the diagonally implicit formula that uses a lower triangular matrix with identical diagonal entries, allowing a maximum of one lower-upper (LU) decomposition per integration stage to be performed will result in substantial computing benefits to the user. The numerical results and plots of accuracy indicate that the ρ -DIBBDF method performs better than the existing fully and diagonally Block Backward Differentiation Formula (BBDF) methods.

The Double Phospho/Dephosphorylation Cycle as a Benchmark to Validate an Effective Taylor Series Method to Integrate Ordinary Differential Equations

“…Numerical simulations of the system (16), quadratized in the form (8) as developed in the previous section, have been performed in the Matlab® suite. Inspired by the parameter choices in [14,17], we consider the following parameters:…”

“…Summing up all left hand sides of the system (16) yields zero, which, on the other hand, is entailed by (14), accounting that M T (the total mass) is constant over time. The algebraic equation ( 14) defines a six-dimensional manifold in IR 7 , invariant with respect to system (16), which means that, if one takes the initial value on this manifold, the evolution of system ( 16) remains on the same manifold at all times. Remark 1.…”

“…Indeed, in enzymatic networks, the binding/unbinding reactions use to occur at a faster rate, thus leading to stiff ODEs, thus characterized by numerical instability when ordinary numerical schemes are employed for their integration. In particular, in the case of stiff equations, ad hoc integration methods, like the one illustrated in [16] for linear multistep methods, can be employed to approximate the solution efficiently. Like any approximations, there are limitations that may render unfeasible its concrete applicability.…”

Approximating non linear higher order ODEs by a three point block algorithm

“…Recent advancement in related area can be found in works such as [14][15][16][17][18][19][20][21][22][23][24][25][26][27] and others.…”

Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

“…However, the well-known problem of such methods is a decrease in numerical stability with an increase in the accuracy order of the applied scheme. This property restricts the application of high-order multistep methods for solving stiff ODE systems, excluding implicit methods, for example, backward differentiation formulas [1][2][3]. Another promising class of numerical integration methods is semi-explicit and semi-implicit integrators [4][5][6].…”