Time-marching algorithms for initial-boundary value problems based upon ''approximate approximations''

New time marching algorithms for solving initial-boundary value problems for semilinear parabolic and hyperbolic equations are described. With respect to the space variable the discretization is based upon a method of ''approximate approximation'' proposed by the second author. We use ''approximate approximations'' of the fourth order. In time the algorithms are finite-difference schemes of either first or second approximation order. At each time step the approximate solution is represented by an explicit analytic formula. The algorithms are stable under mild restrictions to the time step which come from the non-linear part of the equation. Some computational results and hints on crucial implementation issues are provided.