Application of large time step TVD high order scheme to shallow water equations

Renyi Xu, Alistair Borthwick, Bo Xu

Research output: Contribution to journalArticlepeer-review


Numerical modeling of actual river floods faces three challenges related to computational efficiency, accuracy, and the well balanced requirement, all of which are discussed in this paper. Herein, a Large Time Step (LTS) scheme is used to improve efficiency, a high order scheme enhances accuracy, and specific treatment of the bed slope term achieves a well-balanced form of the shallow water equations. The LTS scheme originally proposed by LeVeque in 1998 has led to the development of highly efficient computational solvers of the Shallow Water Equations (SWEs). This paper examines the use of a Total Variation Diminishing (TVD) high order scheme in conjunction with LTS. We first apply the scheme to the solution of the homogeneous 1D SWEs and obtain satisfactory results for three cases, even though small oscillations nevertheless occur when the CFL number is very large. The additional source term makes the issue more complicated and can introduce a spurious flow when the term is not well handled. Many methods have been developed in traditional differential schemes but not all are fit for the TVD-LTS scheme; for example, the method decomposing the source term into simple characteristic waves has proved not feasible. Herein TVD-LTS scheme is implemented for the first time for well-balanced SWEs containing bed-slope source terms. We find that oscillations are not as suppressed as for the homogeneous SWEs when the TVD-LTS scheme is applied to three cases of Step Riemann Problems (SRP) tested for CFL numbers 1 to 10. For free surface flow over a bed hump, the TVD-LTS scheme can only reach a CFL number 4 before the solution breaks down.
Original languageEnglish
Article number1856
Number of pages17
Issue number11
Early online date8 Nov 2022
Publication statusE-pub ahead of print - 8 Nov 2022


  • Shallow water equations
  • Large time step scheme
  • TVD


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