In solving hyperbolic conservation laws of the form \(u_ t+f(u)_ x=0\), it is desirable to have a method that is at least second-order accurate in smooth regions of the flow and that also gives sharp resolution of discontinuities with no spurious oscillations.
The geometric approach, similar to D. van Leer’s MUSCL schemes [J. Comput. Phys. 23, 263-275 (1977; Zbl 0339.76039) and Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method, ibid. 32, 101-136 (1979)], is used to construct a second order accurate generalization of Godunov’s method for solving scalar conservation laws. By making suitable approximations a scheme which is easy to implement is obtained. The entropy condition is investigated from the standpoint of the spreading of rarefaction waves. For Godunov’s method the quantitative information is obtained concerning the rate of spreading which explains the kinks in rarefaction waves often observed at the sonic point.