We present a new nonlinear algorithm for the e cient and accurate solution of isothermal and nonisothermal phase change problems. The method correctly evolves latent heat release in isothermal and nonisothermal phase change, and more importantly, it provides a means for the e cient and accurate coupling between temperature and concentration elds in multi-species nonisothermal phase change. The method rigorously conserves energy both globally and locally. Newton-like super-linear convergence is

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... chieved in the global nonlinear iteration, without the complexity of forming or inverting the Jacobian matrix. This "Jacobian-free" method is a combination of an outer Newton-based iteration and an inner conjugate gradient-like Krylov iteration. The e ects of the Jacobian are probed only through approximate matrix-vector products required in the conjugate gradient-like iteration. The methodology behind the Jacobian-free Newton-Krylov solution method is given in detail. We demonstrate the properties of this method which allow the formulation of an implicit solution algorithm having enthalpy as the dependent v ariable. The performance of the method is demonstrated on phase change problems for a pure material undergoing isothermal solidi cation and a binary eutectic alloy undergoing nonisothermal solidi cation.