The Boussinesq-Peregrine (BP) equations describe shallow water regimes with a time-dependent bathymetry, which can naturally be derived from the water wave theory, [1]. The determination of bottom bathymetry in terms of measurements can be formulated as an optimization PDE-constraint problem. The solution of this, can be done by means of the descent-step approach in which two hyperbolic systems are involved. One of them corresponds to BP, the constraint of the optimization problem and the other to the so-called adjoint system, which is non-conservative and evolves back in time. Classical solution strategies consist mainly of the solution of these two problems by means of conservative schemes in which non-conservative parts are dealt as source terms. It is also usual to use different solvers for each type of equation. Both systems can be putted together in a unified form which results into a non-conservative hyperbolic system, [2]. We profit from the low-discipation scheme FORCE-α, reported in [3], and the unified formulation to build an universal scheme which can be applied to both systems, [4]. The scheme results to be efficient compared with the use of conservative schemes. This does not require any adaptation to solve both forward (BP) and backward (adjoint) evolutions. The approach allows to recover very well both continuous and discontinuous bottom profiles. REFERENCES [1] D. Lannes. The water waves problem, Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013, DOI 10.1090/surv/188. Mathematical analysis and asymptotics. [2] G. I. Montecinos, J. Lopez-Rios, J. H. Ortega and R. Lecaros. A numerical procedure and coupled system formulation for the adjoint approach in hyperbolic PDE-constrained optimization problems. IMA Journal of Applied Mathematics, 84(3): 483-516, 2019. [3] E. Toro, B. Saggiorato, S. Tokareva and A. Hidalgo. Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 2020, 416, 109545. [4] R. Lecaros, J. Lpez-Ros, G. I. Montecinos, E. Zuazua. Optimal control approach for moving bottom detection in one-dimensional shallow waters by surface measurements, Math. Meth. Appl. Sci. 1-32, 2024.