On finite‐difference approximations and entropy conditions for shocks

Abstract

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.

On the link between weighted least-squares and limiters used in higher-order reconstructions for finite volume computations of hyperbolic equations

“…In order to incorporate differences in the influence of each neighbouring point on the gradient (which is usually based on distance from the node), the least squares method needs to be modified. This was achieved by modifying equation (11) with the introduction of a diagonal matrix (called weight matrix) to yield the weighted least squares approximation [7],…”

“…However, classical higher-order schemes, whilst giving high resolutions to discontinuities of the solutions, exhibits spurious oscillations around such locations. Whereas, any monotone scheme [11], that guarantees non-oscillatory solution, can only be first-order accurate when a linear approach is used. Therefore, 'weak monotone concept' [16], using non-linear limiter schemes, is introduced to limit anti-diffusive terms present in the high accuracy schemes [20].…”

Reply [to “Comment on ‘Modeling the magnetosphere for northward interplanetary magnetic field: Effects of electrical resistivity’ by Joachim Raeder”]

“…Thus one strives to develop schemes that are monotone [Godunov, 1959], that is, schemes that do not produce artificial extrema. Godunov [1959], and later Harten et al [1976] and Osher [1983] showed that monotone schemes are at most locally first-order accurate and thus excessively diffusive.…”

Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations

“…On the other hand, monotone schemes for first-order conservation laws (corresponding to A ≡ 0) were first analyzed in [12,13]. Their attractive feature is the convergence to an entropy solution, which remains valid for the application to strongly degenerate parabolic equations.…”