The Random Choice Method

Abstract

This chapter introduces the celebrated random choice method, which has provided the earliest, but still very effective, scheme for constructing globally defined, admissible BV solutions to the Cauchy problem for strictly hyperbolic systems of conservation laws, under initial data with small total variation. The solution is obtained as the limit of a sequence of approximate solutions that do not smear shocks. Solutions to the Riemann problem, discussed at length in Chapter IX, serve as building blocks for constructing the approximate solutions to the Cauchy problem. Striving to preserve the sharpness of shocks may be in conflict with the requirement of consistency of the algorithm. The “randomness” feature of the method is employed in order to strike the delicate balance of safeguarding consistency without smearing the sharpness of propagating shock fronts. At the cost of delineating the global wave pattern, the device of wave tracing, which will be discussed here only briefly, renders the algorithm deterministic. A detailed presentation of the random choice method will be given for systems with characteristic families that are either genuinely nonlinear or linearly degenerate. The case ofmore general systems, which involves substantial technical complication, will be touched on rather briefly here. The chapter will close with a discussion on how the algorithm may be adapted for handling inhomogeneity and source terms encountered in hyperbolic systems of balance laws.