A mass conservative, well balanced, tangency preserving and energy decaying method for the shallow water equations on a sphere

Max Gunzburger, Buyang Li, Jilu Wang, Zongze Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review


A fully discrete surface finite element method is proposed for solving the viscous shallow water equations in a bounded Lipschitz domain on the sphere based on a general triangular mesh. The method consists of a modified Crank–Nicolson method in time and a Galerkin surface finite element method in space for the fluid thickness H and the fluid velocity u. A finite element space tangential to the sphere at all finite element nodes is proposed to approximate the fluid velocity u. The proposed method has second-order accuracy in time and first-order accuracy in space, and preserves mass conservation, well balancedness, tangency of velocity to the sphere, and energy decay. Numerical experiments are presented to illustrate the accuracy of the proposed method and the preservation of the physical properties, including mass conservation, well balancedness, and energy decay. A numerical simulation of ocean mesoscale activity on a circular basin with a continental shelf is provided.

Original languageEnglish
Article number111067
Pages (from-to)1-18
Number of pages18
JournalJournal of Computational Physics
Publication statusPublished - 15 May 2022


  • Energy decay
  • Mass conservation
  • Shallow water equations on a sphere
  • Well balancedness

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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