Taylor Grid Discretization for Evolution Partial Differential Equations
Abstract: We propose a high order discretization scheme for evolution PDE's, both linear and non-linear. Assuming that the equation and its solutions are sufficiently smooth, we reduce the initial PDE to a relatively compact system of ordinary differential equations. The unknowns of this system are the (time depending) Taylor coefficients of the solution with respect to the space variable, up to a certain fixed order N, computed at a certain fixed space-grid.
Algebraically, the evolution equations for these Taylor coefficients are obtained by the "Jet extension" of the original equation, combined with the Generalized Hermite Interpolation. The last is used to express the derivatives of the orders higher that N, which naturally arise in the Jet extension, through the Taylor coefficients up to order N at the neighboring grid-points.
The main feature of the proposed discretization scheme is that its order of accuracy (expressed as the power of the grid size h) is significantly higher than the order $N$ of the Taylor polynomials explicitly used. This is achieved via the use of the high order Generalized Hermite Interpolation with the neighboring grid-points.