lax_friedrichs_flux

Family: finite_volume
Grid family: cartesian
Kind: scheme
Accuracy: O(h)
Applies to: flux(q, c), dim=x
Rule file: discretizations/finite_volume/lax_friedrichs_flux.json
Tags: #finite-volume #lax-friedrichs #upwind #flux #monotone #tvd #first-order

Continuous form

lax_friedrichs_flux discretizes the conservative 1D advective face flux on a uniform Cartesian grid for linear advection $f(q) = c,q$,

$$F(x_{i+1/2},\,t)\;=\;f\big(q(x_{i+1/2},\,t)\big),$$

with a face-staggered Courant number $c_{i+1/2} = u_{i+1/2},\Delta t/\Delta x$. The Lax & Friedrichs (1954) numerical flux is

$$F_{i+1/2}\;=\;\tfrac{c}{2}\,(q_i + q_{i+1})\;-\;\tfrac{|c|}{2}\,(q_{i+1} - q_i),$$

which, collecting terms, is algebraically identical to first-order upwinding:

$$F_{i+1/2}\;=\;\frac{c + |c|}{2}\,q_i \;+\; \frac{c - |c|}{2}\,q_{i+1}\;=\;\max(c,0)\,q_i \;+\; \min(c,0)\,q_{i+1}.$$

Upwind cell selection is encoded directly in the abs op — no caller-side branching on $\operatorname{sign}(c)$ is required.

Stencil

A two-point face stencil straddling the interface $F_{i+1/2}$ (the right cell of the interface is offset 0):

selector kind	axis	offset	coeff
`cartesian`	`$x`	0	`(c +
`cartesian`	`$x`	1	`(c −

Combined as +. For $c > 0$ the offset-0 coefficient is $c$ and the offset-1 coefficient is $0$ (pure upwind from the left cell); for $c < 0$ the weights swap (upwind from the right cell). The closed flux_form AST in the JSON states the same identity directly as

$$F_{i+1/2}\;=\;\frac{c + |c|}{2}\,q_i \;+\; \frac{c - |c|}{2}\,q_{i+1}.$$

Properties

Monotone and TVD — the scheme is dissipative by construction; it adds the first-order numerical diffusion characteristic of upwinding and cannot produce new extrema.
The face-Courant $c is a per-face binding supplied by the caller (analogous to the slope-ratio $r in flux_limiter_minmod), not a stencil-offset selector.
Retained as a debugging / oracle scheme. Production transport should use the higher-order PPM or WENO rules (flux_1d_ppm, weno5_advection).

Composition

The caller computes the face-staggered Courant $c_{i+1/2} = u_{i+1/2},\Delta t/\Delta x$ and applies this rule at every interface; the cell-centred tendency follows from the FV divergence

$$T_i\;=\;-\,\frac{F_{i+1/2} - F_{i-1/2}}{h}$$

on a uniform grid (or composes with divergence_arakawa_c on a C-grid). It is the first-order sibling of flux_1d_ppm: same op = flux shape and per-face binding contract, with the LF abs-based first-order upwind in place of PPM’s higher-order reconstruction + Courant-fraction flux integral.

Convergence

The rule’s Layer-B convergence fixture lives under discretizations/finite_volume/lax_friedrichs_flux/fixtures/convergence/. Like flux_1d_ppm, a flux-form transport sweep needs the per-face Courant binding ($c) and a Layer-B′ MMS-transport runner (numerical flux + FV divergence + time stepping) to measure empirical order on a smooth profile; until those harness features land the convergence fixture is suppressed and the algebraic upwind identity above is exercised inline.

Reference

Lax, P. and K. O. Friedrichs (1954). “Weak solutions of nonlinear hyperbolic equations and their numerical computation.” Comm. Pure Appl. Math. 7(1), 159–193 — original LF flux.
First-order reduction: for linear advection with face-staggered Courant $c$, $F_{i+1/2} = \max(c,0),q_i + \min(c,0),q_{i+1}$.
Higher-order sibling: flux_1d_ppm (PPM flux-form transport).