Test case following H84. Typical flow patterns for χ = 1.5 and χ = 21 are shown in panels a and b, respectively, in (dimensionless) absolute velocity (color coded) with velocity arrows superimposed. Horizontally averaged vertical profiles of v_rms, convective (F_C), kinetic (F_K), and radiative (F_R) fluxes are given in panels c–f. Colors denote solutions obtained with the well-balanced scheme for χ = 1.5 (dark green) and χ = 21 (light green), with the well-balanced scheme but central differences for S^E and χ = 21 (yellow), and with the standard scheme for χ = 21 (red). Solid lines are time averages, dashed lines are temporal variability (±1 standard deviation, where distinguishable from the mean). Panel g: time integrated gain of energy via S^E if the scheme is not well balanced in energy (red dashed and blue solid) or if the standard scheme (purple dashed and cyan solid) is used for χ = 1.5 (dashed lines) and χ = 21 (solid lines, scaled by a factor of 30).