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Discrete advection–diffusion equations on graphs: Maximum principle and finite volumes

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  • Hošek, Radim
  • Volek, Jonáš

Abstract

We study an initial value problem for explicit and implicit difference advection–diffusion equations on graphs. Problems on both finite and infinite graphs are considered. We analyze the existence and uniqueness of solutions. Interestingly, we show that there exist infinitely many solutions to implicit problems on infinite graphs similarly as in the case of continuous or lattice diffusion equations on infinite spatial domains. The main part of the paper is devoted to maximum principles. Firstly, we establish discrete maximum principles for equations on graphs. Then we show that finite volume numerical schemes for advection–diffusion PDEs in any dimension can be reformulated as equations on graphs and consequently, we use this relation to verify maximum principles for corresponding numerical solutions.

Suggested Citation

  • Hošek, Radim & Volek, Jonáš, 2019. "Discrete advection–diffusion equations on graphs: Maximum principle and finite volumes," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 630-644.
    • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:630-644
    DOI: 10.1016/j.amc.2019.06.014
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    References listed on IDEAS

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    1. Petr Stehlík & Jonáš Volek, 2015. "Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-13, September.
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