Stage-parallel preconditioners for implicit Runge-Kutta methods of arbitrarily high order, linear problems

• Fully implicit Runge–Kutta methods offer the possibility to use high order accurate time discretization to match space discretization accuracy, an issue of significant importance for many large scale problems of current interest, where we may have fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this work, we consider strongly A-stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures. For the arising large algebraic systems we introduce efficient preconditioners, that (1) use only real arithmetic, (2) demonstrate robustness with respect to problem and discretization parameters, and (3) allow for fully stage-parallel solution. The preconditioners are based on the observation that the lower-triangular part of the coefficient matrices in the Butcher tableau has larger in magnitude values, compared to the corresponding strictly upper-triangular part. We analyze the spectrum of the corresponding preconditioned systems and illustrate their performance with numerical experiments. Even though the observation has been made some time ago, its impact on constructing stage-parallel preconditioners has not yet been done and its systematic study constitutes the novelty of this article.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

fully stage-parallel preconditioning

implicit Runge–Kutta methods

parallelization

Radau quadrature

Beräkningsvetenskap med inriktning mot numerisk analys

Scientific Computing with specialization in Numerical Analysis

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