Riemann solvers and high-resolution reconstruction

For Burgers’ law with flux one half u squared, a left state greater than a right state forms an entropy shock with speed(U_L+U_R)/2. A smaller left state forms a rarefaction. The interface flux is the physical flux of the entropy solution sampled at similarity coordinate x/t=0. In a rarefaction spanning zero, that sampled value is zero, not an average of the endpoint fluxes. In a stationary shock, both traces have the same physical flux, so the flux is still well defined even though the point value at the shock is not unique.

For a convex scalar flux, the Godunov flux can be written as the minimum of the flux over the state interval when the left state is no larger than the right, and the maximum over the reversed interval otherwise. This formula incorporates the entropy solution. It is not a componentwise recipe for a general hyperbolic system, where characteristic waves interact. Approximate solvers use wave-speed estimates and numerical dissipation; their consistency, admissibility and positivity properties must be checked for the particular system and parameter choices.

On a uniform grid, reconstruct cell i as U_i+s_i(x−x_i), which preserves its cell average because the linear correction has zero mean on the cell. Minmod chooses the common-sign slope of smaller magnitude from the left and right difference quotients, and returns zero when their signs disagree or one is zero. The face values are U_i plus or minus half the cell width times that slope. A reconstruction built from nodal values rather than cell averages would need a different conservation argument.

For smooth monotone data away from extrema, left and right slopes differ by a small Taylor error, so minmod can retain useful slope information and a second-order spatial reconstruction. At a smooth extremum those slopes have opposite signs, forcing a zero slope and locally reducing formal accuracy. This is the price of a simple total-variation-diminishing limiter. Statements about uniform high order at every smooth point therefore need more refined limiting strategies and hypotheses; they do not follow just from writing a piecewise linear polynomial.

A static piecewise linear reconstruction gives instantaneous interface states. To obtain a second-order fully discrete method, time evolution or an appropriate multistage integration must also achieve the required order. Feeding reconstructed states into a first-order time formula does not automatically yield second-order space-time accuracy. Likewise, a strong-stability-preserving Runge–Kutta theorem inherits a forward-Euler stability property only under a proven forward-Euler step restriction and the method’s SSP coefficient; it is not an unrestricted guarantee for any reconstruction.

For scalar TVD claims, identify the flux, limiter form, mesh and Courant interval used in the proof. A condition for one flux-limiter formulation is not a general constant for all MUSCL or system solvers. Near discontinuities, limiting may intentionally suppress high-order terms; compare conservation and entropy behavior as well as smooth-solution error. The calculations below isolate reconstructed traces and the entropy Godunov flux, making clear which parts of a completed high-resolution update have and have not yet been constructed.