Summary.
We study the \(L^1\)-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 \(\epsilon\) denote the `small scale' of such approximations (– the viscosity amplitude \(\epsilon\), the spatial grad-size \(\Delta x\), 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 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 \(L^1\)-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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Received April 20, 1998 / Revised version received November 8, 1999 / Published online August 24, 2000
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Lin, CT., Tadmor, E. $L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions. Numer. Math. 87, 701–735 (2001). https://doi.org/10.1007/PL00005430
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DOI: https://doi.org/10.1007/PL00005430