  In this work, the Haar wavelet method for the coupled non-linear transient
PDE-ODEs system with the Neumann boundary condition has been proposed. The
capability of the method in handling multiple jump discontinuities in the
coefficients and parameters of the transient and coupled PDE-ODEs system is
brought out through a series of 1D/2D/3D test problems. The method is easy to
implement, computationally efficient, and compares well with conventional
methods like the finite element method. Convergence analysis of the method has
been carried out and apriori error is found to be exponential order i.e.
||u-u_H||_X<= o(2^j ). ILU-GMRES is found to better accelerate the numerical
solution convergence than other Krylov solvers for the class of transient
non-linear PDE-ODE system.
