Finite-volume fluxes and conservation

Consider a one-dimensional density q measured as mass per unit length, moving at constant positive speed a. Integrating the conservation law over a fixed cell relates the rate of cell mass to flux through its two endpoints. Define qj as a cell average, not a point value. An explicit finite-volume step replaces each time-averaged endpoint flux with a numerical approximation. For positive-speed first-order upwinding, the interface takes the density from its left, upstream cell.

The resulting update is a convex combination of the old cell and its upstream neighbor when the Courant number lies between zero and one. This preserves nonnegative averages from nonnegative input and limits this scheme's explicit step. It also smooths a transported discontinuity, producing numerical diffusion. A sharp-looking profile is therefore not the only measure of quality: conservation, positivity, error in transported position and convergence under refinement all describe different aspects of the approximation.

Sum cell masses after the update. Each internal interface enters twice with opposite signs, leaving only the outer boundary fluxes. Periodicity identifies the two outer fluxes, so the total mass is unchanged up to arithmetic error. At an open boundary the mass may legitimately change. Treating any mass loss as a numerical bug would confuse a physical outflow with a violation of the discrete balance.

For positive advection speed, an inflow condition is supplied at the left boundary while information exits at the right. Prescribing arbitrary independent data at both ends can overconstrain the transport problem. A wall, an inflow and a periodic wrap represent different experiments. Record flux conventions and compare the observed domain mass change with the integrated boundary transfer. A shared interface update also makes debugging local conservation possible before a full simulation has reached an interesting flow pattern.

For nonlinear conservation laws, discontinuities are interpreted through integral balances rather than ordinary derivatives at the jump. Consistent conservative methods have an important relation to weak solutions when a suitable convergent limit exists. That statement does not prove the sequence converges, remains positive or satisfies the physical entropy condition. Numerical fluxes, limiting and time integration must address those issues. A nonconservative discretization can even propagate a shock at the wrong speed while behaving acceptably on smooth data.

The relevant entropy inequality is a mathematical admissibility condition for the chosen conservation law, not an assertion that every computed cell has a thermodynamic entropy variable. The Markov lab offers a small example of nonnegative weights that preserve a total. Its two-state probabilities are exact matrix updates, not cell averages on an advecting mesh. It can distinguish conservation from convergence to equilibrium, but it has no physical interface flux, shock solver or boundary inflow.