Convergence of numerical schemes involving powers of the Dirac delta function

Abstract

Developments of the Hull elastoplastic numerical method lead to nonconservative versions, which in the case of shock waves involve multiplications of distributions of the type of powers of the Dirac delta function. In the one dimensional case of the shock wave equation u_t + uu_x = 0, the numerical solutions will converge to the solution of a different equation, if the convergence and the latter equation are considered within the nonlinear theory of generalized functions introduced recently by the second author. The study of this phenomenon, presented here in one of its relevant particular cases, offers for the first time a rigorous understanding of important similar situations encountered in industrial applications, when numerical solutions may show either agreement with, or deviations from the expected solutions.