Numerical analysis in singularly perturbed boundary value problems modelling heat transfer processes
We construct a finite difference method for boundary value problems modelling heat and mass transfer for fast-running processes. The dimensionless form of the equation in these problems is singularly perturbed, i.e., the highest derivatives are multiplied by a parameter ε2 which can take any values from the interval (0,1]. The equation involves concentrated sources; the boundary conditions are mixed. As is known, classical numerical methods lead us to large errors that can exceed many times the exact solution for small ε; a similar problem occurs if we are to find the normalized flux, i.e., the gradient multiplied by ε. New special schemes are constructed to converge uniformly with respect to the parameter. The errors in the discrete solution and in the computed fluxes are independent of the parameter. The new schemes can be applied to the analysis of heat exchange in metal working by hot die-forming or for plastic shear.
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