Two Approximations of Solutions of Hamilton-Jacobi Equations* By M. G. Crandall** and P. L. Lions

Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. The associated initial-value problems almost never have global-time classical solutions, and one must deal with suitable generalized solutions. The correct class of generalized solutions has only recently been established by the authors. This article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates. Analogous results are proved by similar means for the method of vanishing viscosity. Introduction. The main results of this paper concern the approximation of solutions of the Cauchy problem for first-order partial differential equations of Hamilton-Jacobi type. Most of the presentation here will be in the context of problems of the form j du/dt + H(Du) = 0 inR^xtO.oo), \u(x,0) = u0(x) inR", where H G C(RN) (the continuous functions on R"), u0 G BUQR") (the bounded and uniformly continuous functions on RN), and Du = (ux,...,ux ) is the spatial gradient of u. The problem (IVP) is technically simpler than the "general case" in which the Hamiltonian H may depend on x, t and u as well as Du, and we prefer to keep the ideas clear and constants simple by dealing primarily with (IVP). (See the comments in Section 4 regarding more general equations.) Two sorts of approximations of (IVP) will be considered here—finite difference schemes and the method of vanishing viscosity. Before describing these approximations, we briefly review some basic facts concerning (IVP). Analysis by the method of chacteristics shows that if H and u0 are smooth and u0 is compactly supported, then (IVP) will typically have a unique C2 solution u on some maximal time interval 0 < t < T for which limr T T u(x, t) exists uniformly, but this limiting function is not continuously differentiable. Thus Du " becomes discontinuous" at t = T (or "shocks form"). If one insists upon a solution of (IVP) which is defined for all / > 0, it is therefore necessary to deal with functions which are not Received September 28, 1982. 1980 Mathematics Subject Classification. Primary 65M15, 65M10.