Journal Article10.1007/BF00178771
Implicit-explicit methods for reaction-diffusion problems in pattern formation
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TL;DR: This work analyzes the performance of several of the best known linear multistep IMEX schemes for reaction-diffusion problems in pattern formation and finds that first order accurate schemes, as well as schemes which produce only a weak decay of high frequency spatial error may yield plausible results which are nonetheless qualitatively incorrect.
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Abstract: Reaction-diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems. To approximate the corresponding spatially discretized models, an explicit scheme can be used for the reaction term and an implicit scheme for the diffusion term. Such implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods in fluid flow problems.
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Citations
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
TL;DR: Runge-Kutta-based IMEX schemes are developed that have better stability regions than the best known IMEX multistep schemes over a wide parameter range.
1.3K
Implicit-explicit methods for time-dependent partial differential equations
TL;DR: This work systematically analyze the performance of implicit-explicit IMEX schemes, propose improved new schemes, and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.
1.1K
Fourth-Order Time-Stepping for Stiff PDEs
TL;DR: A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators.
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
TL;DR: Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations and results for the fifth-order method are disappointing, but both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods.
849
Multiphysics simulations: Challenges and opportunities
David E. Keyes,Lois Curfman McInnes,Carol S. Woodward,William Gropp,Eric Myra,Michael Pernice,John B. Bell,Jed Brown,Alain Clo,Jeffrey M. Connors,Emil M. Constantinescu,Donald Estep,Katherine J. Evans,Charbel Farhat,Ammar Hakim,Glenn E. Hammond,Glen A. Hansen,Judith Hill,Tobin Isaac,Xiangmin Jiao,Kirk E. Jordan,Dinesh K. Kaushik,Efthimios Kaxiras,Alice Koniges,Kihwan Lee,P. Aaron Lott,Qiming Lu,John H. Magerlein,Reed M. Maxwell,Michael McCourt,Miriam Mehl,Roger P. Pawlowski,Amanda Randles,Daniel R. Reynolds,Béatrice Rivière,Ulrich Rüde,Timothy D. Scheibe,John N. Shadid,Brendan Sheehan,Mark S. Shephard,Andrew R. Siegel,Barry Smith,Xian-Zhu Tang,Cian R. Wilson,Barbara Wohlmuth +44 more
- 01 Feb 2013
TL;DR: This study considers multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity, and “architectural’ includes both software and hardware environments.
355
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