On Positivity of the Local IterationModified Time Integration Scheme

• Published in: Lobachevskii journal of mathematics Vol. 46; no. 1; pp. 31 - 42
• Main Authors: Botchev, Mikhail A., Zhukov, Victor T.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.01.2025; Springer Nature B.V
• Subjects: Algebra; Analysis; Chebyshev approximation; Geometry; Linear systems; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Parabolic differential equations; Partial differential equations; Polynomials; Probability Theory and Stochastic Processes; Runge-Kutta method; Stability; Time integration; local iteration modified (LIM) scheme; exponential time integration; positivity; explicit stabilized Runge–Kutta schemes; Runge–Kutta–Chebyshev methods; monotonicity; 2010 Mathematics Subject Classification: 65M20, 65L04, 65L20
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224608403

Local iteration modified (LIM) scheme is an explicit time integration method where accuracy and stability for large time steps are provided by Chebyshev polynomial iterations. The LIM scheme possesses two characteristic features: (1) the Chebyshev polynomial in the LIM scheme is constructed to give a higher accuracy for the important low frequency solution modes, (2) the number of Chebyshev iterations is chosen depending on time step size, to have a stable time integration (rather than to solve arising linear system as done in implicit schemes). The LIM scheme can be seen as a stabilized explicit Runge–Kutta method where Chebyshev iterations play the role of internal stages which guarantee stability. For parabolic PDEs the number of Chebyshev iterations in the LIM scheme is inversely proportional to the spatial grid size  . An important property of the LIM scheme which makes it unique in the class of explicit iterative time integration schemes, is positivity. In this work we thoroughly study this property of the scheme. Up to now, positivity of the LIM scheme has been shown experimentally in numerical tests as well as, partially, theoretically. Our aim here is to provide further theoretical and practical insight in the positivity properties of the scheme.