A fully-implicit numerical method based upon adaptively refined meshes for the\ud simulation of binary alloy solidification in 2D is presented. In addition we combine a\ud second-order fully-implicit time discretisation scheme with variable steps size control\ud to obtain an adaptive time and space discretisation method. The superiority of this\ud method, compared to widely used fully-explicit methods, with respect to CPU time\ud and accuracy, is shown. Due to the high non-linearity of the governing equations\ud a robust and fast solver for systems of nonlinear algebraic equations is needed to\ud solve the intermediate approximations per time step. We use a nonlinear multigrid\ud solver which shows almost h-independent convergence behaviour
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