Foliation structures and global flow dynamics of scalar hyperbolic conservation laws on manifolds: I. geometry-compatible fluxes and numerical validation on sphere and torus

This work develops a theoretical and computational framework for scalar hyperbolic conservation laws (sHCL) posed on closed, two-dimensional (2D) manifolds. The key contribution is a geometry-compatible (GC) flux formulation based on prescribed flux directional vectors, which guarantees consistency between surface divergence and manifold geometry. This construction induces a natural foliation: the manifold decomposes into leaves along which the 2D sHCL reduces to a family of one-dimensional (1D) HCL problems. Global flow dynamics then arise as the collective evolution of these 1D leaf-wise solutions. Part I emphasizes this GC flux formulation, its foliation-based reduction, and the validation of the 2D-1D correspondence. Numerical simulations, performed using a cp-WENO scheme that combines the Closest Point Method (CPM) embedding with high-order WENO discretization, are primarily employed to substantiate the theoretical analysis. Experiments with the inviscid Burgers’ equation illustrate the framework on both the sphere and torus. On the sphere, shocks and rarefaction waves evolve along circular leaves, and the longest leaf emerges as an asymptotic separatrix dividing large-scale rotational patterns. On the torus, the foliation depends critically on the flux vector: in the degenerate case, singular circular leaves act as invariant barriers separating clockwise and counterclockwise flows; in the generic case, isolated singular points anchor rotational structures and organize nonlinear wave interactions. These results confirm that nontrivial geometric features (regular and singular leaves, barriers, and separatrices) govern the global flow on manifolds. More broadly, the study establishes that 2D sHCL on manifolds can be rigorously analyzed and faithfully approximated as a collection of 1D sHCL defined along foliated leaves. Part II will extend the framework to more general fluxes and provide full details of the numerical methodology.