A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds

Yinghua Wang, Bao-Shan Wang, Leevan Ling, Wai Sun Don*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review


A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, and closed manifolds (surface PDEs or SPDEs in R2 and R3). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers’ equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers’ equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers’ equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.

Original languageEnglish
Article number84
Number of pages26
JournalJournal of Scientific Computing
Issue number3
Early online date9 Nov 2022
Publication statusPublished - Dec 2022

Scopus Subject Areas

  • Theoretical Computer Science
  • Software
  • Numerical Analysis
  • Engineering(all)
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Closest point embedding
  • Surface PDEs
  • Central finite volume
  • ENO interpolation
  • Burgers’
  • Traffic flow
  • Buckley-Leverett


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