Traveltime computation is widely used in seismic modeling, imaging, and velocity analysis. The two most commonly used methods are ray tracing and numerical solutions to the eikonal equation. Eikonal solvers are fast and robust but are limited to computing only the first-arrival traveltimes. Ray tracing can compute multiple arrivals but lacks the robustness of eikonal solvers. We propose a robust and complete method of traveltime computation. It is based on a system of partial differential equations, which is equivalent to the eikonal equation but formulated in the ray-coordinates system. We use a first-order discretization scheme that is interpreted very simply in terms of the Huygens's principle. Our explicit finite-difference solution to the eikonal equation solved in the ray-coordinates system delivers both computational speed and stability since we use more than one point on the current wavefront at every time step. The finite-difference method has proven to be a robust alternative to conventional ray tracing, while being faster and having a better ability to handle rough velocity media and penetrate shadow zones.