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Additive Runge-Kutta schemes for convection-diffusion-reaction equations. (English) Zbl 1013.65103
Summary: Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when $$N$$ different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, ($$N=2$$), additive Runge-Kutta (ARK$$_2$$) methods from third- to fifth-order are presented that allow for integration of stiff terms by an $$L$$-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK).
Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK$$_2$$ methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, $$z^{[I]}\to -\infty$$, and retain high stability efficiency in the absence of stiffness, $$z^{[I]}\to 0$$. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK$$_2$$ error terms and Butcher coefficient magnitudes as well as maximize conservation properties.
Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK$$_2$$ methods.

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35B25 Singular perturbations in context of PDEs 35K57 Reaction-diffusion equations
##### Software:
OEIS; Mathematica; KELLEY; TR-BDF2; RODAS
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