A Stable Domain Decomposition Technique for Advection–Diffusion Problems

Oskar Ålund, Jan Nordström

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


The use of implicit methods for numerical time integration typically generates very large systems of equations, often too large to fit in memory. To address this it is necessary to investigate ways to reduce the sizes of the involved linear systems. We describe a domain decomposition approach for the advection–diffusion equation, based on the Summation-by-Parts technique in both time and space. The domain is partitioned into non-overlapping subdomains. A linear system consisting only of interface components is isolated by solving independent subdomain-sized problems. The full solution is then computed in terms of the interface components. The Summation-by-Parts technique provides a solid theoretical framework in which we can mimic the continuous energy method, allowing us to prove both stability and invertibility of the scheme. In a numerical study we show that single-domain implementations of Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to compute solutions for grid resolutions that cannot be handled efficiently using a single-domain formulation. An order of magnitude speed-up is observed, both compared to a single-domain formulation and to explicit Runge–Kutta time integration.

Original languageEnglish
Pages (from-to)755-774
Number of pages20
JournalJournal of Scientific Computing
Issue number2
Publication statusPublished - 1 Nov 2018
Externally publishedYes


  • Domain decomposition
  • Finite difference methods
  • Partial differential equations
  • Stability
  • Summation-by-Parts

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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