A. HARTEN, J. M. HYMAN AND P. D. LAX with Appendix by B. KEYFITZ This paper is dedicated to Robert D. Richtmyer in recognition of his many important contributions to the numerical solutions of partial differential equations and of his great influence on scientific computing.
AbstractWeak solutions of hyperbolic conservation laws are not uniquely determined by their initial values; an entropy condition is needed to pick out the physically relevant solution. The question arises whether finite-difference approximations converge to this particular solution. It is shown in this paper that in the case of a single conservation law, monotone schemes, when convergent, always converge to the physically relevant solution. Numerical examples show that this is not always the case with non-monotone schemes, such as the Lax-Wendroff scheme. @ 1976 by John Wiley & Sons, Inc.298 A. HARTEN, J. M. HYMAN AND P. D . LAXwhich asserts that u is constant along the characteristic curvesThe constancy of u along the characteristics combined with (1.2b) implies that the characteristics are straight lines. Their slope, however, depends upon the solution and therefore they may intersect, and where they do, no continuous solution can exist. To get existence in the large, i.e., for all time, we admit weak solutions which satisfy an integral version of (1. l), m
[ w t u + w x f ( u ) ] d x d t + w ( x , O ) + ( x ) d x = Ofor every smooth test function w ( x , t ) of compact support.(see [ 101) that across the line of discontinuity the Rankine-Hugoniot relation If u is a piecewise continuous weak solution, then it follows from (1.3) holds, where S is the speed of propagation of the discontinuity, and uL and uR are the states on the left and on the right of the discontinuity, respectively. The class of all weak solutions is too wide in the sense that there is no uniqueness for the initial value problem, and an additional principle is needed for determining a physically relevant solution. Usually this principle identifies the physically relevant solution as a limit of solutions with some dissipation, namelyOleinik [ 151 has shown that discontinuities of such admissible solutions can be characterized by the following condition: for all u between uL and uR ; this is called the entropy condition, or Condition E. Oleinik has shown, see 11151, that weak solutions satisfying Condition E are uniquely determined by their initial data. Another elegant proof is given in c71.
