Abstract

This work presents a novel class of fully-discrete and semi-discrete Lagrangian-Eulerian schemes for three-dimensional scalar and systems of hyperbolic conservation laws on structured cubical and tetrahedral meshes. Built upon the mathematical framework of no-flow curves, surfaces, and manifolds, this Riemann-solver-free approach reinterprets the evolution of hyperbolic fluxes as a vector field problem. This framework yields an effective weak CFL-type stability condition and natural positivity, completely bypassing the need for spectral information, exact/approximate eigenvalues, or Jacobian constructions. Consequently, time-consuming field-by-field decompositions are avoided, retaining the simplicity of lower-dimensional formulations. The dual-step evolution-remap strategy eliminates moving meshes while ensuring local conservation. We demonstrate the method’s versatility and performance through high-performance computing (MPI) simulations of compressible Euler flows, the Orszag-Tang MHD problem (maintaining $\nabla \cdot B=0$ without additional constraints), 2D shallow water equations with variable and discontinuous topography, and non-strictly hyperbolic three-phase flows with resonance points. Strong scaling metrics verify the theory and confirm computational efficiency.