A simple moving mesh method for blowup problems

We develop a simple and efficient adaptive mesh generator for time‐dependent partial differential equations with singular solutions in two dimensional spaces. The mesh generator is based on minimizing the sum of two diagonal lengths in each cell. We also add second order difference terms to obtain smoother and more orthogonal mesh. The method is successfully applied to the nonlinear heat equations with blowup solutions. We can obtain a solution with an amplitude of 10¹⁵ at the peak and the mesh difference of 10⁻¹⁶ near the peak. We also discuss nonlinear heat equations whose solutions blow up at space infinity and whose blowup time is given.