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A classical way of finding the harmonic map is to minimize the harmonic energy by the time evolution of the solution of a nonlinear heat diffusion equation. To arrive at the desired harmonic map, which is a steady-state of this equation, we propose an efficient quasi-implicit Euler method and analyze its convergence under some simplifications. If the initial map is not close to the steady-state solution, it is difficult to find the stability region of the time steps. To remedy this drawback, we propose a two-phase approach for the quasi-implicit Euler method. In order to accelerate the convergence, a variant time step scheme and a heuristic method to determine an initial time step are developed. Numerical results clearly demonstrate that the proposed method achieves high performance for computing the spherical conformal and Riemann mappings.

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