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This thesis by Cristina La Cognata investigates numerical methods for solving discontinuous and nonlinear systems of partial differential equations using high-order finite differences in the summation-by-parts (SBP) framework. The work focuses on ensuring stability and conservation properties through weak imposition of interface and boundary conditions with the simultaneous approximation term (SAT) technique. The research is divided into three parts, addressing simulations of discontinuous linear advection problems, vorticity-stream function formulation for inviscid fluids, and the incompressible Navier-Stokes equations. The study aims to improve long-term numerical stability and accuracy in applications such as climate modeling, fluid dynamics, and ocean circulation predictions. The intended audience includes researchers and practitioners in computational mathematics and engineering, particularly those interested in numerical analysis and differential equations.

Numerical analysis. --- Differential equations, Partial. --- Numerical analysis --- Differential equations, Partial