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High Order Numerical Discretization for Hamilton-Jacobi Equations on Triangular Meshes

Summary: High Order Numerical Discretization for Hamilton-Jacobi
Equations on Triangular Meshes 
Steeve Augoula y and Rémi Abgrall z
Mathématiques Appliquées de Bordeaux, Université de Bordeaux I
351 Cours de la Libération, 33 405 Talence Cedex, France
In this paper we construct several numerical approximations for rst order Hamilton-Jacobi
equations on triangular meshes. We show that, thanks to a ltering procedure, the high order
versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are
based on the rst order Lax-Friedrichs scheme [2] which is improved here adjusting the dissipation
term. The resulting rst order scheme is -monotonic (we explain the expression in the paper) and
converges to the viscosity solution as O(
t) for the L 1 -norm. The rst high order method is
directly inspired by the ENO philosophy in the sense where we use the monotonic Lax-Friedrichs
Hamiltonian to reconstruct our numerical solutions. The second high order method combines a
spatial high order discretization with the classical high order Runge-Kutta algorithm for the time
discretization. Numerical experiments are performed for general Hamiltonians and L 1 , L 2 and
L 1 -errors with convergence rates calculated in one and two space dimensions show the k-th order
rate when piecewise polynomial of degree k functions are used, measured in L 1 -norm.


Source: Abgrall, Rémi - Institut de Mathematiques de Bordeaux, Université Bordeaux


Collections: Mathematics