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We develop an adjoint approach for recovering the topographical function included in the source term of one-dimensional hyperbolic balance laws. We focus on a specific system, namely the shallow water equations, in an effort to recover the riverbed topography. The novelty of this work is the ability to robustly recover the bottom topography using only noisy boundary data from one measurement event and the inclusion of two regularization terms in the iterative update scheme. The adjoint scheme is determined from a linearization of the forward system and is used to compute the gradient of a cost function. The bottom topography function is recovered through an iterative process given by a three-operator splitting method which allows the feasibility to include two regularization terms. Numerous numerical tests demonstrate the robustness of the method regardless of the choice of initial guess and in the presence of discontinuities in the solution of the forward problem.
The regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations. In a recent work, the first two authors proposed a so-called H
We carry out the group classification of the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The equivalence group of this class is found by the algebraic
We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with
We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equ
In this paper we analyze the stability of equilibrium manifolds of hyperbolic shallow water moment equations. Shallow water moment equations describe shallow flows for complex velocity profiles which vary in vertical direction and the models can be s