Positivity-preserving methods for ordinary differential equations
- Publisher:
- EDP Sciences
- Publication Type:
- Journal Article
- Citation:
- ESAIM Mathematical Modelling and Numerical Analysis, 2022, 56, (6), pp. 1843-1870
- Issue Date:
- 2022-11-01
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Many important applications are modelled by differential equations with positive solutions However it remains an outstanding open problem to develop numerical methods that are both i of a high order of accuracy and ii capable of preserving positivity It is known that the two main families of numerical methods Rungea Kutta methods and multistep methods face an order barrier If they preserve positivity then they are constrained to low accuracy They cannot be better than first order We propose novel methods that overcome this barrier second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions Our methods apply to a large class of differential equations that have a special graph Laplacian structure which we elucidate The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric This algebraic structure arises naturally in many important applications where positivity is required We showcase our new methods on applications where standard high order methods fail to preserve positivity including infectious diseases Markov processes master equations and chemical reactions The authors Published by EDP Sciences SMAI 2022
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