Some well-balanced Finite Volume solvers for Shallow Water Equations

The accurate computation of stationary solutions of hyperbolic systems with source terms has been found in the past years as closely related to the accurate computation of transient solutions. Numerical solvers that do not solve steady solutions up to second order, at least, yield transient solutions that present large errors that grow as time increases, unacceptable from the physical point of view.

These solvers are called “well-balanced”, as the flow and source terms must balance each other, up to high precision, for steady solutions.This talk focus on the systematic derivation of numerical schemes satisfying this property.

Starting from standard Finite Volume solvers that can be written in viscous form, we shall derive associated well-balanced solvers in a specific way: These compute all steady solutions up to second order, in all the domain but on a subdomain whose measure vanishes as the grid size goes to zero.

We shall apply these solvers to Shallow Water equations with variable bottom and friction effects. In particular we shall present a numerical simulation of the toxic waste due to the breaking of Aznalcóllar mine pond in 1998.