A general, fully spectral weighted residual method for solution of initial value partial differential equations is presented. All time, spatial and physical parameter domains are represented by Chebyshev series enabling global semi-analytical, rather than purely numerical, solutions. The method avoids time step limitations. The spectral coefficients are determined by iterative solution of a system of algebraic equations, for which a globally convergent root solver has been developed. Accuracy is controlled by the number of included Chebyshev modes and by the use of spatial subdomains. It is shown by example and by comparisons with the explicit Lax-Wendroff and semi-implicit Crank-Nicholson methods that the method may be used for accurate and efficient solution of nonlinear initial value problems in fluid mechanics and magnetohydrodynamics.