Conservative Schemes
Conservative schemes are written as differences of numerical fluxes so that conserved quantities change only through domain boundaries.
Flux-difference form
A numerical scheme for a conservation law is conservative if the update to each cell is expressed as the difference of numerical fluxes at its faces: the change in a cell equals flux in minus flux out. When a shared face uses the same numerical flux for both adjacent cells, the flux leaving one cell exactly enters its neighbor. Summing over all cells, the interior contributions telescope and cancel, so the total conserved quantity changes only through fluxes at the domain boundary.
This discrete conservation mirrors the physical conservation law exactly, independent of grid resolution or solution smoothness, which is why it is a foundational requirement for transport and fluid solvers.
Why it matters for shocks
Beyond bookkeeping, conservation determines correctness at discontinuities. The Lax-Wendroff theorem states that if a consistent conservative scheme converges, its limit is a weak solution of the conservation law with the right jump conditions and shock speeds. Non-conservative discretizations can appear reasonable yet converge to solutions with shocks moving at the wrong speed, an error that mesh refinement does not cure.
Building conservative schemes
Finite volume methods are conservative by construction because they discretize the integral form directly. Finite difference schemes can be made conservative by writing them in flux-difference form. Care is needed at mesh interfaces, coarse-fine boundaries in AMR, and coordinate singularities, where matching fluxes preserves the telescoping cancellation.
- Update = flux in minus flux out for each cell
- Shared face flux guarantees telescoping cancellation
- Lax-Wendroff: convergence yields correct weak solutions
- Flux matching required at mesh and refinement interfaces
Conservation is non-negotiable for credible long-time transport in fusion neutronics, edge-plasma, and MHD modeling, where drift in a conserved budget signals a numerical rather than a physical effect.