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Preprint 2006-048

Convergence rate of monotone numerical schemes for Hamilton–Jacobi equations with weak boundary conditions

Knut Waagan

Abstract: We study a class of monotone numerical schemes for time-dependent Hamilton–Jacobi equations with weak Dirichlet boundary conditions. We get a convergence rate of 1/2 under some usual assumptions on the data, plus an extra assumption on the Hamiltonian H(Du,x) at the boundary ∂Ω. More specifically the mapping pH(p,x) must satisfy a monotonicity condition for all p in a certain subset of Rn given by Ω. This condition allows the use of the interior subsolution conditions at the boundary in the comparison arguments. We also prove a comparison result and Lipschitz regularity of the exact solution. As an example we construct a Godunov type scheme that can handle the weakened boundary conditions.

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Knut Waagan,
Publishing information:
Updated 2007-03-09.
Submitted by:
; 2006-12-06.