fronttracking

Front tracking is a numerical method for solving hyperbolic partial differential equations, particularly conservation laws, by explicitly tracking discontinuities in the solution, such as shocks and material interfaces. Instead of diffusing sharp features, fronts are represented as moving lower-dimensional structures embedded in the computational domain. The evolution of these fronts is governed by the governing equations and by local Riemann problems at each front, which determine the speeds and strengths of the fronts. When fronts collide or interact, a local Riemann problem is solved to produce new fronts, and the front representation is updated accordingly. Solutions can be advanced with either exact or approximate Riemann solvers, and the method aims to maintain sharp resolution of discontinuities.

Front tracking is often used in one- or quasi-one-dimensional problems where discontinuities play a central role, such as gas dynamics, shallow water equations, and combustion. In higher dimensions, fronts become curves or surfaces, and their management involves splitting, merging, and reparametrization, increasing computational complexity. Variants may combine front tracking with level-set techniques or particle-based representations to handle topology changes or complex geometries. The approach requires careful treatment of front curvature, weak solutions, and consistency with conservation laws.

Strengths of front tracking include accurate capture of sharp interfaces and reduced numerical diffusion compared with traditional grid-based schemes. Limitations include difficulty handling large numbers of frontal interactions, topological changes, and scalability in higher dimensions. As a specialized tool, front tracking is most effective in problems where the correct representation of moving discontinuities is essential.