Abstract
This paper is concerned with monotone (time-explicit) finite difference scheme associated with first order Hamilton–Jacobi equations posed on a junction. It extends the scheme introduced by Costeseque et al. (Numer Math 129(3):405–447, 2015) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton–Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive some optimal error estimates of in L^{infty }_{text {loc}} for junction conditions of optimal-control type.
Highlights
This paper is concerned with numerical approximation of first order Hamilton–Jacobi equations posed on a junction, that is to say a network made of one node and a finite number of edges
It is understood that general junction conditions reduce to special ones of optimal-control type [25]
Numerical schemes for Hamilton–Jacobi equations on networks The discretization of viscosity solutions of Hamilton–Jacobi equations posed on networks has been studied in a few papers only
Summary
This paper is concerned with numerical approximation of first order Hamilton–Jacobi equations posed on a junction, that is to say a network made of one node and a finite number of edges. Our second and main result is an error estimate in the style of Crandall–Lions [17] in the case of flux-limited junction conditions It is explained in [17] that the proof of the comparison principle between sub- and super-solutions of the continuous Hamilton– Jacobi equation can be adapted in order to derive error estimates between the numerical solution associated with monotone (stable and consistent) schemes and the continuous solution. In the case of a continuous equation, the comparison principle is proved thanks to the technique of doubling variables; it relies on the classical penalisation term ε−1|x − y|2 Such a penalisation procedure is known to fail in general if the equation is posed on a junction, because the equation is discontinuous in the space variable; it is explained in [25] that it has to be replaced with a vertex test function.
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