In this work, we present a practical example of co-design between numerical analysis and high performance computing, applied to compressible fluid mechanics. We consider a legacy numerical method based on the Lagrange-Remap solver for the compressible Euler equations, and use tools from analytical performance modeling -- the roofline and ECM models -- to quantitatively understand its behavior on recent multicore CPUs, and extract precise knowledge of its computational bottlenecks. This analysis prompts us to propose a new numerical method, called Lagrange-Flux. Experimental results backed by our theoretical understanding show that this new method yields an algorithm that is more scalable, more computationally efficient, and at the same time retains the good numerical properties of the original Lagrange-Remap solver. This is a joint work with Florian De Vuyst, Thibault Gasc, Mathieu Peybernes, Renaud Motte.