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In this work, the Haar wavelet method for the coupled non-linear transientPDE-ODEs system with the Neumann boundary condition has been proposed. Thecapability of the method in handling multiple jump discontinuities in thecoefficients and parameters of the transient and coupled PDE-ODEs system isbrought out through a series of 1D/2D/3D test problems. The method is easy toimplement, computationally efficient, and compares well with conventionalmethods like the finite element method. Convergence analysis of the method hasbeen 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 numericalsolution convergence than other Krylov solvers for the class of transientnon-linear PDE-ODE system.

An efficient Haar wavelet method for the coupled non-linear transient PDE-ODEs system with discontinuous coefficients