Double scale method for solving nonlinear systems involving multiplications of distributions
In this paper, we develop a double scale numerical method which may be of great significance in computing discontinuous solutions of certain system of PDE's arising in application. It is of basic importance, especially in industrial applications, to study the solutions which represent shock waves. Then there appear "multiplications of distributions" in the form Y (.) delta of a product of the Heaviside and the Dirac functions; their intuitive treatment may lead to "ambiguous results". To remove this ambiguity, J. F. Colombeau developed a mathematical theory by introducing the differential algebra G(R x R+) of generalized functions where discontinuous functions are involved through classes of C-infinity functions. The aim of this paper is to show how this method can be implemented in new numerical methods. We apply our apply to a system modeling elasticity and a system modeling gas dynamics. We observe a complete agreement between the theory (algebraic solutions of Riemann problems) and the numerical results.
double scale method generalized functions nonconservative system numerical scheme