Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

We consider explicit two-step peer methods for the solution of nonstiff differential systems. By an additional condition a subclass of optimally zero-stable methods is identified that is superconvergent of order p = s + 1, where s is the number of stages. The new condition allows us to reduce the number of coefficients in a numerical search for good methods. We present methods with 4-7 stages which are tested in FORTRAN90 and compared with DOPRI5 and DOP853. The results confirm the high potential of the new class of methods.

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“…Now let the inequality be satisfied for j ≤ m − 1, ε j ≤ j−1 l=1 (1 + 2ch l )dh s+1 . Then, (12) gives…”

Section: So the Error Recursion Becomesmentioning

confidence: 99%

“…In numerical tests on parallel computers they were rather efficient, e.g. [14,12], and also in sequential computing environments they were competitive with standard codes [7]. In [15] corresponding explicit methods were introduced and tests with parallel explicit two-step peer methods were performed in [11].…”

Section: Introductionmentioning

confidence: 99%

Superconvergent explicit two-step peer methods

Weiner

Schmitt

Podhaisky

et al. 2009

Journal of Computational and Applied Mathematics

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“…The purpose of our paper is to design an efficient and accurate adaptive ODE solver for stiff and very stiff problems, which arise in various areas of applied research, as those listed above. This new computational technique is grounded in recently designed implicit (and linearly implicit) peer methods [2,37,39,40,51]. Our choice is because of two reasons.…”

Section: Introduction Ordinary Differential Equations (Odes) Of the mentioning

confidence: 99%

A Singly Diagonally Implicit Two-Step Peer Triple with Global Error Control for Stiff Ordinary Differential Equations

Kulikov¹,

Weiner²

2015

SIAM J. Sci. Comput.

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This paper elaborates a new numerical method for treating stiff ordinary differential equations (ODEs) which constitute a basic simulation tool in many areas of study. For its efficiency, this technique is grounded in singly diagonally implicit two-step peer schemes designed recently. The latter means that linear systems with the same coefficient matrix are solved to advance a step of the method and to evaluate its global error as well. Moreover, in contrast to many other numerical methods with global error control, we monitor and regulate the scaled global error, which corresponds better with practitioners' needs. In other words, we present and study one peer triple which is able to solve stiff problems and to ensure that the true error of the derived numerical solution does not exceed the user-supplied bound at the same time. This implies that local and global error estimation and control mechanisms are also discussed here. Finally, efficiency of our numerical tool is checked in comparison to efficiency of stiff built-in MATLAB ODE solvers on various test problems.Key words. stiff ordinary differential equations, singly diagonally implicit peer methods, absolute and scaled local and global error estimations, automatic local and global error controls

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“…Later the methods were generalized in several directions, in [10] implicit parallel methods were considered, in [7] sequential linearly-implicit methods were introduced. In [9] linearly-implicit parallel peer methods were combined with the full orthogonalization method (FOM) and were used for the solution of large stiff systems of ordinary differential equations resulting from the semidiscretization of partial differential equations with the method of lines (MOL). In this paper we will combine these approaches and consider s-stage sequential implicit peer methods for large stiff systems.…”

Section: Introductionmentioning

confidence: 99%

Implicit peer methods for large stiff ODE systems

Beck

Weiner

Podhaisky

et al. 2011

J. Appl. Math. Comput.

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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.

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Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

Cited by 41 publications

References 14 publications

Superconvergent explicit two-step peer methods

Superconvergent explicit two-step peer methods

A Singly Diagonally Implicit Two-Step Peer Triple with Global Error Control for Stiff Ordinary Differential Equations

Implicit peer methods for large stiff ODE systems

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