Computation of nonclassical solutions to Hamilton-Jacobi problems

This paper is devoted to the construction of numerical methods for the appr oximation of nonclassical solutions to multidimensional Hamilton{Jacobi equ ations for both scalar and vectorial problems. Recent theoretical results h ave yielded existence of solutions in many cases for which the usual viscos ity approach was ill-suited or not applicable. The selection criterion used here is based on a viscoelasticity/capillarity approach, common in solid m echanics. Numerical methods adapted to this framework are built. Consistenc y of the model equation with the given selection criterion is essential. It is achieved here through the use of high-order finite difference schemes. By considering applications to potential well problems, the convergence of the methods are investigated.