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Mathematics of Computation

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Stability and boundedness in the numerical solution of initial value problems

Author: M. N. Spijker
Journal: Math. Comp. 86 (2017), 2777-2798
MSC (2010): Primary 65L20, 65M12; Secondary 65L05, 65L06, 65M20
Published electronically: March 3, 2017
MathSciNet review: 3667024
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This paper concerns the theoretical analysis of step-by-step methods for solving initial value problems in ordinary and partial differential equations.

The main theorem of the paper answers a natural question arising in the linear stability analysis of such methods. It guarantees a (strong) version of numerical stability—under a stepsize restriction related to the stability region of the numerical method and to a circle condition on the differential equation.

The theorem also settles an open question related to the properties total-variation-diminishing, strong-stability-preserving, monotonic and (total- variation-)bounded. Under a monotonicity condition on the forward Euler method, the theorem specifies a stepsize condition guaranteeing boundedness for linear problems.

The main theorem is illustrated by applying it to linear multistep methods. For important classes of these methods, conclusions are thus obtained which supplement earlier results in the literature.

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Additional Information

M. N. Spijker
Affiliation: Department of Mathematics, University of Leiden, PO Box 9512, NL-2300-RA Leiden, Nederland
MR Author ID: 165560

Keywords: Numerical stability, circle condition, total-variation-diminishing (TVD), strong-stability-preserving (SSP), monotonicity, total-variation-bounded (TVB), boundedness, method of lines (MOL), linear multistep method
Received by editor(s): January 28, 2016
Received by editor(s) in revised form: June 2, 2016
Published electronically: March 3, 2017
Article copyright: © Copyright 2017 by the author