  Different relaxation approximations to partial differential equations,
including conservation laws, Hamilton-Jacobi equations, convection-diffusion
problems, gas dynamics problems, have been recently proposed. The present paper
focuses onto diffusive relaxed schemes for the numerical approximation of
nonlinear reaction diffusion equations. We choose here a nonstandard relaxation
scheme that allow the treatment of diffusion equations in their nonconservative
form. A comparison with the traditional approach in the case of conservative
equations is also included.
  High order methods are obtained by coupling ENO and WENO schemes for space
discretization with IMEX schemes for time integration, where the implicit part
can be explicitly solved at a linear cost. To illustrate the high accuracy and
good properties of the proposed numerical schemes, also in the degenerate case,
we consider various examples in one and two dimensions: the Fisher-Kolmogoroff
equation, the porous-Fisher equation and the porous medium equation with strong
absorption. Moreover we show a test on a system of PDEs that describe an
ecological model for the dispersal and settling of animal populations.
