L1-Stability and error estimates for approximate Hamilton-Jacobi solutions

We study theL1-stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let denote the ‘small scale’ of such approximations ( – the viscosity amplitude , the spatial grad-size ∆x, etc.), then our L1-error estimates are of O( ), and are sharper than the classical L∞-results of order one half,O(√ ). The main building blocks of our theory are the notions of the semi-concave stability condition and L1-measure of the truncation error. The whole theory could be viewed as amultidimensional extension of theLip′-stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain L1-bounds on their associated truncation errors; L1-convergence of order one then follows. Second-order (central) Godunov-type schemes are