Implicit peer methods for large stiff ODE systems

Abstract

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi's method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p = s in the multi-implicit case and order p = s − 1 in the singly-implicit case with arbitrary step sizes and s ≤ 5. Numerical tests in MATLAB for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.

New third- and fourth-order singly diagonally implicit two-step peer triples with local and global error controls for solving stiff ordinary differential equations

“…Their code outperforms the stiff second-order Matlab solver ODE23s. It is grounded in s-stage singly diagonally implicit two-step peer schemes [15]. When applied on a variable mesh w τ := {t k+1 = t k + τ k , k = 0, 1, .…”

“…The order conditions (6) are obtained by the Taylor expansion of the left-hand side of formula (5) (given by (4)) around the mesh node t k [15,17]. The order conditions (6) are a powerful means for constructing implicit two-step peer formulas of the form (2) and of consistency order p in practice.…”

“…where R is a constant fixed independently of meshes w τ ∈ W Ω ω (t 0 , t end ) [15,17]. We specify that the first condition in (7) is needed for zero-stability (8), whereas the other one ensures the correctness of our global error estimation and control mechanism presented in Section 2.3.…”

“…Besides, the A-stability study of the discussed peer schemes (as well as similar studies in other general linear methods [1,7]) is performed for their fixed-stepsize variants. So, following [15,17], we apply method (3) with τ k = τ and B(k) = B and arrive at the recursion X k = M (z)X k−1 where the stability matrix…”

Application of approximate matrix factorization to high order linearly implicit Runge–Kutta methods

“…But their methods are limited to the low or medium precision level, and no generalization to higher order is supplied. In [5] Beck et al compared the efficiency of AMF versus Krylov based approaches to the solution of linear systems in the context of Newton iterations arising in Radau [13] and Peer [6] integration methods. These methods avoid the issue of order degradation, through the use of integration schemes in which the Jacobian of the spatial discretization does not appear explicitly.…”