Abstract
We propose a technique for investigating stability properties like positivity and forward invariance of an interval for method-of-lines discretizations, and apply the technique to study positivity preservation for a class of TVD semi-discretizations of 1D scalar hyperbolic conservation laws. This technique is a generalization of the approach suggested in Khalsaraei (J Comput Appl Math 235(1): 137–143, 2010). We give more relaxed conditions on the time-step for positivity preservation for slope-limited semi-discretizations integrated in time with explicit Runge–Kutta methods. We show that the step-size restrictions derived are sharp in a certain sense, and that many higher-order explicit Runge–Kutta methods, including the classical 4th-order method and all non-confluent methods with a negative Butcher coefficient, cannot generally maintain positivity for these semi-discretizations under any positive step size. We also apply the proposed technique to centered finite difference discretizations of scalar hyperbolic and parabolic problems.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 244-266 |
| Number of pages | 23 |
| Journal | Journal of Scientific Computing |
| Volume | 74 |
| Issue number | 1 |
| DOIs | |
| State | Published - Apr 19 2017 |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledgements: This work was supported by the King Abdullah University of Science and Technology (KAUST), 4700 Thuwal, 23955-6900, Saudi Arabia. The first author was also supported by the Tempus Public Foundation. The third author was also supported by the Department of Numerical Analysis, Eötvös Loránd University, and the Department of Differential Equations, Budapest University of Technology and Economics, Hungary.