A well-balanced reconstruction with bounded velocities for the shallow water equations by convex combination

06/21/2021
by   Edward W. G. Skevington, et al.
0

Finite volume schemes for hyperbolic balance laws require a piecewise polynomial reconstruction of the cell averaged values, and a reconstruction is termed `well-balanced' if it is able to simulate steady states at higher order than time evolving states. For the shallow water system this involves reconstructing in surface elevation, to which modifications must be made as the fluid depth becomes small to ensure positivity, and for many reconstruction schemes a modification of the inertial field is also required to ensure the velocities are bounded. We propose here a reconstruction based on a convex combination of surface and depth reconstructions which ensures that the depth increases with the cell average depth. We also discuss how, for cells that are much shallower than their neighbours, reducing the variation in the reconstructed flux yields bounds on the velocities. This approach is generalisable to high order schemes, problems in multiple spacial dimensions, and to more complicated systems of equations. We present reconstructions and associated technical results for three systems, the standard shallow water equations, shallow water in a channel of varying width, and a shallow water model of a particle driven current. Positivity preserving time stepping is also discussed.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

06/21/2021

A Well Balanced Reconstruction with Bounded Velocities and Low-Oscillation Slow Shocks for the Shallow Water Equations

Many numerical schemes for hyperbolic systems require a piecewise polyno...
04/09/2020

A two-dimensional high-order well-balanced scheme for the shallow water equations with topography and Manning friction

We develop a two-dimensional high-order numerical scheme that exactly pr...
09/05/2019

Finding optimal hull shapes for fast vertical penetration into water

A new approach for the supercavitating hull optimization was proposed, w...
06/05/2020

Ensuring 'well-balanced' shallow water flows via a discontinuous Galerkin finite element method: issues at lowest order

The discontinuous Galerkin finite element method (DGFEM) developed by Rh...
03/26/2021

Three dimensional higher-order raypath separation in a shallow-water waveguide

Separating raypaths in a multipath shallow-water environment is a challe...
12/17/2021

Barrier Simulations with Shallow Water Equations using the State Redistribution Method

The representation of small scale barriers, such as sea-walls, in coasta...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.