Abstract:We study the $L^1$-stability and error estimates of general approximate solutions of the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain {\em 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 approximate solutions: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let $\epsilon$ denote the 'small scale' of such approximations (\dd the viscosity amplitude $\epsilon$, the spatial grad-size $\dx$, etc.), then our $L^1$-error estimates are of ${\cal O}(\epsilon)$, and are sharper than the classical $L^\infty$-results of order one half, ${\cal O}(\sqrt{\epsilon})$.
The main building blocks of our theory are the notions of the semi-concave stability condition and $L^1$-measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the $Lip^\prime$-stability theory for one-dimensional nonlinear conservation laws by Nessyahu Tadmor and Tassa \cite{Ta91,NeTa92,NeTaTa}, when we regard the viscosity solution of H-J equation as the primitive of the entropy solution for the corresponding conservation law.
In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a {\em 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 {\em nonlinear} projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain $L^1$-bounds on their associated truncation errors; $L^1$-convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our $L^1$-theory.
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